9.7: Graph Quadratic Functions Using Properties
By the end of this section, you will be able to:
- Recognize the graph of a quadratic function
- Find the axis of symmetry and vertex of a parabola
- Find the intercepts of a parabola
- Graph quadratic functions using properties
- Solve maximum and minimum applications
Before you get started, take this readiness quiz.
-
Graph the function \(f(x)=x^{2}\) by plotting points.
If you missed this problem, review Example 3.54. -
Solve: \(2 x^{2}+3 x-2=0\).
If you missed this problem, review Example 6.45. -
Evaluate \(-\frac{b}{2 a}\) when \(a=3\) and \(b=-6\).
If you missed this problem, review Example 1.21.
Recognize the Graph of a Quadratic Function
Previously we very briefly looked at the function \(f(x)=x^{2}\), which we called the square function. It was one of the first non-linear functions we looked at. Now we will graph functions of the form \(f(x)=a x^{2}+b x+c\) if \(a \neq 0\). We call this kind of function a quadratic function.
A quadratic function , where \(a, b\), and \(c\) are real numbers and \(a≠0\), is a function of the form
\(f(x)=a x^{2}+b x+c\)
We graphed the quadratic function \(f(x)=x^{2}\) by plotting points.
Every quadratic function has a graph that looks like this. We call this figure a parabola . Let’s practice graphing a parabola by plotting a few points.
Graph: \(f(x)=x^{2}-1\).
Solution :
We will graph the function by plotting points.
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Choose integer values for \(x\),
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Plot the points, and then connect
them with a smooth curve. The result will be the graph of the function \(f(x)=x^{2}-1\). |
Graph \(f(x)=-x^{2}\).
- Answer
Graph \(f(x)=x^{2}-1\).
- Answer
All graphs of quadratic functions of the form \(f(x)=a x^{2}+b x+c\) are parabolas that open upward or downward. See Figure 9.6.6
Notice that the only difference in the two functions is the negative sign before the quadratic term (\(x^{2}\) in the equation of the graph in Figure 9.6.6 ). When the quadratic term, is positive, the parabola opens upward, and when the quadratic term is negative, the parabola opens downward.
Parabola Orientation
For the graph of the quadratic function \(f(x)=a x^{2}+b x+c\), if
Determine whether each parabola opens upward or downward:
- \(f(x)=-3 x^{2}+2 x-4\)
- \(f(x)=6 x^{2}+7 x-9\)
Solution :
a. Find the value of \(a\).
Since the \(a\) is negative, the parabola will open downward.
b. Find the value of \(a\).
Since the \(a\) is positive, the parabola will open upward.
Determine whether the graph of each function is a parabola that opens upward or downward:
- \(f(x)=2 x^{2}+5 x-2\)
- \(f(x)=-3 x^{2}-4 x+7\)
- Answer
-
- up
- down
Determine whether the graph of each function is a parabola that opens upward or downward:
- \(f(x)=-2 x^{2}-2 x-3\)
- \(f(x)=5 x^{2}-2 x-1\)
- Answer
-
- down
- up
Find the Axis of Symmetry and Vertex of a Parabola
Look again at Figure 9.6.10 . Do you see that we could fold each parabola in half and then one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.
We show the same two graphs again with the axis of symmetry.
The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of \(f(x)=a x^{2}+b x+c\) is \(x=-\frac{b}{2 a}\).
So to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula \(x=-\frac{b}{2 a}\).
Notice that these are the equations of the dashed blue lines on the graphs.
The point on the parabola that is the lowest (parabola opens up), or the highest (parabola opens down), lies on the axis of symmetry. This point is called the vertex of the parabola.
We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means its
\(x\)-coordinate is \(-\frac{b}{2 a}\). To find the \(y\)-coordinate of the vertex we substitute the value of the \(x\)-coordinate into the quadratic function.
Axis of Symmetry and Vertex of a Parabola
The graph of the function \(f(x)=a x^{2}+b x+c\) is a parabola where:
- the axis of symmetry is the vertical line \(x=-\frac{b}{2 a}\).
- the vertex is a point on the axis of symmetry, so its \(x\)-coordinate is \(-\frac{b}{2 a}\)
- the \(y\)-coordinate of the vertex is found by substituting \(x=-\frac{b}{2 a}\) into the quadratic equation.
For the graph of \(f(x)=3 x^{2}-6 x+2\) find:
- the axis of symmetry
- the vertex
Solution :
a.
| The axis of symmetry is the vertical line \(x=-\frac{b}{2 a}\). | |
| Substitute the values \(a,b\) into the equation. | \(x=-\frac{-6}{2 \cdot 3}\) |
| Simplify. | \(x=1\) |
| The axis of symmetry is the line \(x=1\). |
b.
| \(f(x)=3 x^{2}-6 x+2\) | |
| The vertex is a point on the line of symmetry, so its \(x\)-coordinate will be \(x=1\). Find \(f(1)\). | |
| Simplify. | |
| The result is the \(y\)-coordinate. | \(f(1)=-1\) |
| The vertex is \((1,-1)\). |
For the graph of \(f(x)=2 x^{2}-8 x+1\) find:
- the axis of symmetry
- the vertex
- Answer
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- \(x=2\)
- \((2,-7)\)
For the graph of \(f(x)=2 x^{2}-4 x-3\) find:
- the axis of symmetry
- the vertex
- Answer
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- \(x=1\)
- \((1,-5)\)
Find the Intercepts of a Parabola
When we graphed linear equations, we often used the \(x\)- and \(y\)-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.
Remember, at the \(y\)-intercept the value of \(x\) is zero. So to find the \(y\)-intercept, we substitute \(x=0\) into the function.
Let’s find the \(y\)-intercepts of the two parabolas shown in Figure 9.6.20 .
An \(x\)-intercept results when the value of \(f(x)\) is zero. To find an \(x\)-intercept, we let \(f(x)=0\). In other words, we will need to solve the equation \(0=a x^{2}+b x+c\) for \(x\).
\(\begin{aligned} f(x) &=a x^{2}+b x+c \\ 0 &=a x^{2}+b x+c \end{aligned}\)
Solving quadratic equations like this is exactly what we have done earlier in this chapter!
We can now find the \(x\)-intercepts of the two parabolas we looked at. First we will find the \(x\)-intercepts of the parabola whose function is \(f(x)=x^{2}+4 x+3\).
| \(f(x)=x^{2}+4 x+3\) | |
| Let \(f(x)=0\). | \(\color{red}0\color{black}=x^{2}+4 x+3\) |
| Factor. | \(0=(x+1)(x+3)\) |
| Use the Zero Product Property. | \(x+1=0 \quad x+3=0\) |
| Solve. | \(x=-1 \quad x=-3\) |
| The \(x\)-intercepts are \((-1,0)\) and \((-3,0)\). |
Now we will find the \(x\)-intercepts of the parabola whose function is \(f(x)=-x^{2}+4 x+3\).
| \(f(x)=-x^{2}+4 x+3\) | |
| Let \(f(x)=0\). | \(\color{red}0 \color{black}=-x^{2}+4 x+3\) |
| This quadratic does not factor, so we use the Quadratic Formula. | \(x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\) |
| \(a=-1, b=4, c=3\) | \(x=\frac{-4 \pm \sqrt{4^{2}-4(-1)(3)}}{2(-1)}\) |
| Simplify. | \(x=\frac{-4 \pm \sqrt{28}}{-2}\) |
| \(x=\frac{-4 \pm 2 \sqrt{7}}{-2}\) | |
| \(x=\frac{-2(2 \pm \sqrt{7})}{-2}\) | |
| \(x=2 \pm \sqrt{7}\) | |
| The \(x\)-intercepts are \((2+\sqrt{7}, 0)\) and \((2-\sqrt{7}, 0)\). |
We will use the decimal approximations of the \(x\)-intercepts, so that we can locate these points on the graph,
\((2+\sqrt{7}, 0) \approx(4.6,0) \quad(2-\sqrt{7}, 0) \approx(-0.6,0)\)
Do these results agree with our graphs? See Figure 9.6.34
Find the Intercepts of a Parabola
To find the intercepts of a parabola whose function is \(f(x)=a x^{2}+b x+c\):
\(y\)-intercept
Let \(x=0\) and solve for \(f(x)\).
\(x\)-intercepts
Let \(f(x)=0\) and solve for \(x\)
Find the intercepts of the parabola whose function is \(f(x)=x^{2}-2 x-8\).
Solution :
| To find the \(y\)-intercept, let \(x=0\) and solve for \(f(x)\). | \(f(x)=x^{2}-2 x-8\) |
| \(f(0)=\color{red}0\color{black}^{2}-2 \cdot \color{red}0 \color{black}-8\) | |
| \(f(0)=-8\) | |
| When \(x=0\), then \(f(0)=-8\). The \(y\)-intercept is the point \((0,-8)\). | |
| To find the \(x\)-intercept, let \(f(x)=0\) and solve for \(x\). | \(f(x)=x^{2}-2 x-8\) |
| \(0=x^{2}-2 x-8\) | |
| Solve by factoring. | \(0=(x-4)(x+2)\) |
| \(0=x-4 \quad 0=x+2\) | |
| \(4=x \quad-2=x\) | |
| When \(f(x)=0\), then \(x=4\) or \(x=-2\). The \(x\)-intercepts are the points \((4,0)\) and \((-2,0)\). |
Find the intercepts of the parabola whose function is \(f(x)=x^{2}+2 x-8\).
- Answer
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\(y\) -intercept: \((0,-8) x\) -intercepts \((-4,0),(2,0)\)
Find the intercepts of the parabola whose function is \(f(x)=x^{2}-4 x-12\).
- Answer
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\(y\) -intercept: \((0,-12) x\) -intercepts \((-2,0),(6,0)\)
In this chapter, we have been solving quadratic equations of the form \(a x^{2}+b x+c=0\). We solved for \(x\) and the results were the solutions to the equation.
We are now looking at quadratic functions of the form \(f(x)=a x^{2}+b x+c\). The graphs of these functions are parabolas. The \(x\) - intercepts of the parabolas occur where \(f(x)=0\).
For example:
Quadratic equation
\(\begin{aligned}x^{2}-2 x-15 & =0\quad \text{Let}\:f(x)=0 \\ (x-5)(x+3) &=0 \\ x-5=0\:\:x+3 & =0 \\ x=5\:\:\:x&=-3\end{aligned}\)
Quadratic function
\(\begin{aligned} f(x) &=x^{2}-2 x-15 \\ 0 &=x^{2}-2 x-15 \\ 0 &=(x-5)(x+3) \\ x-5 &=0 \quad x+3=0 \\ x &=5 \quad x=-3 \\(5,0) & \text { and }(-3,0) \\& x\text { -intercepts } \end{aligned}\)
The solutions of the quadratic function are the \(x\) values of the \(x\) - intercepts.
Earlier, we saw that quadratic equations have \(2, 1\), or \(0\) solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the functions give the \(x\)-intercepts of the graphs, the number of \(x\)-intercepts is the same as the number of solutions.
Previously, we used the discriminant to determine the number of solutions of a quadratic function of the form \(a x^{2}+b x+c=0\). Now we can use the discriminant to tell us how many \(x\)-intercepts there are on the graph.
Before you to find the values of the \(x\)-intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.
Find the intercepts of the parabola for the function \(f(x)=5 x^{2}+x+4\).
Solution :
| To find the \(y\)-intercept, let \(x=0\) and solve for \(f(x)\). | |
| When \(x=0\), then \(f(0)=4\). The \(y\)-intercept is the point \((0,4)\). | |
| To find the \(x\)-intercept, let \(f(x)=0\) and solve for \(x\). | |
| Find the value of the discriminant to predict the number of solutions which is also the number of \(x\)-intercepts. | |
| \(\begin{array}{c}{b^{2}-4 a c} \\ {1^{2}-4 \cdot 5 \cdot 4} \\ {1-80} \\ {-79}\end{array}\) | |
|
Since the value of the discriminant is negative, there is no real solution to the equation. There are no \(x\)-intercepts. |
Find the intercepts of the parabola whose function is \(f(x)=3 x^{2}+4 x+4\).
- Answer
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\(y\)-intercept: \((0,4)\) no \(x\)-intercept
Find the intercepts of the parabola whose function is \(f(x)=x^{2}-4 x-5\)
- Answer
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\(y\)-intercept: \((0,-5)\) \(x\)-intercepts \((-1,0),(5,0)\)
Graph Quadratic Functions Using Properties
Now we have all the pieces we need in order to graph a quadratic function. We just need to put them together. In the next example we will see how to do this.
Graph \(f(x)=x^{2}-6x+8\) by using its properties.
Solution :
| Step 1 : Determine whether the parabola opens upward or downward. |
Look \(a\) in the equation \(f(x)=x^{2}-6x+8\) Since \(a\) is positive, the parabola opens upward. |
\(f(x)=x^{2}-6x+8\) \(\color{red}{a=1, b=-6, c=8}\) The parabola opens upward. |
| Step 2 : Find the axis of symmetry. |
\(f(x)=x^{2}-6x+8\) The axis of symmetry is the line \(x=-\frac{b}{2 a}\). |
Axis of Symmetry \(x=-\frac{b}{2 a}\) \(x=-\frac{(-6)}{2 \cdot 1}\) \(x=3\) The axis of symmetry is the line \(x=3\). |
| Step 3 : Find the vertex. | The vertex is on the axis of symmetry. Substitute \(x=3\) into the function. |
Vertex \(f(x)=x^{2}-6x+8\) \(f(3)=(\color{red}{3}\color{black}{)}^{2}-6(\color{red}{3}\color{black}{)}+8\) \(f(3)=-1\) The vertex is \((3,-1)\). |
| Step 4 : Find the \(y\)-intercept. Find the point symmetric to the \(y\)-intercept across the axis of symmetry. |
We find \(f(0)\). We use the axis of symmetry to find a point symmetric to the \(y\)-intercept. The \(y\)-intercept is \(3\) units left of the axis of symmetry, \(x=3\). A point \(3\) units to the right of the axis of symmetry has \(x=6\). |
\(y\)-intercept \(f(x)=x^{2}-6 x+8\) \(f(0)=(\color{red}{0}\color{black}{)}^{2}-6(\color{red}{0}\color{black}{)}+8\) \(f(0)=8\) The \(y\)-intercept is \((0,8)\). Point symmetric to \(y\)-intercept: The point is \((6,8)\). |
| Step 5 : Find the \(x\)-intercepts. Find additional points if needed. |
We solve \(f(x)=0\). We can solve this quadratic equation by factoring. |
\(x\)-intercepts \(f(x)=x^{2}-6 x+8\) \(\color{red}{0}\color{black}{=}x^{2}-6x+8\) \(\color{red}{0}\color{black}{=}(x-2)(x-4)\) \(x=2 or x=4\) The \(x\)-intercepts are \((2,0)\) and \((4,0)\). |
| Step 6 : Graph the parabola. | We graph the vertex, intercepts, and the point symmetric to the \(y\)-intercept. We connect these \(5\) points to sketch the parabola. |
Graph \(f(x)=x^{2}+2x-8\) by using its properties.
- Answer
Graph \(f(x)=x^{2}-8x+12\) by using its properties.
- Answer
We list the steps to take in order to graph a quadratic function here.
To Graph a Quadratic Function Using Properties
- Determine whether the parabola opens upward or downward.
- Find the equation of the axis of symmetry.
- Find the vertex.
- Find the \(y\)-intercept. Find the point symmetric to the \(y\)-intercept across the axis of symmetry.
- Find the \(x\)-intercepts. Find additional points if needed.
- Graph the parabola.
We were able to find the \(x\)-intercepts in the last example by factoring. We find the \(x\)-intercepts in the next example by factoring, too.
Graph \(f(x)=-x^{2}+6 x-9\) by using its properties.
Solution :
| Since \(a\) is \(-1\), the parabola opens downward. | |
| To find the equation of the axis of symmetry, use \(x=-\frac{b}{2 a}\). | \(x=-\frac{b}{2 a}\) |
| \(x=-\frac{6}{2(-1)}\) | |
| \(x=3\) | |
|
The axis of symmetry is \(x=3\). The vertex is on the line \(x=3\). |
|
| Find \(f(3)\). | \(f(x)=-x^{2}+6 x-9\) |
| \(f(3)=-9+18-9\) | |
| \(f(3)=0\) | |
| The vertex is \((3,0)\). | |
| The \(y\)-intercept occurs when \(x=0\). Find \(f(0)\). | \(f(x)=-x^{2}+6 x-9\) |
| Substitute \(x=0\). | |
| Simplify. | \(f(0)=-9\) |
| The point \((0,-9)\) is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is \((6,-9)\). | |
| Point symmetric to the \(y\)-intercept is \((6,-9)\) | |
| The \(x\)-intercept occurs when \(f(x)=0\). | |
| Find \(f(x)=0\). | |
| Factor the GCF. | |
| Factor the trinomial. | |
| Solve for \(x\). | |
| Connect the points to graph the parabola. |
Graph \(f(x)=3 x^{2}+12 x-12\) by using its properties.
- Answer
Graph \(f(x)=4 x^{2}+24 x+36\) by using its properties.
- Answer
For the graph of \(f(x)=-x^{2}+6 x-9\), the vertex and the \(x\)-intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation \(0=-x^{2}+6x-9\) is \(0\), so there is only one solution. That means there is only one \(x\)-intercept, and it is the vertex of the parabola.
How many \(x\)-intercepts would you expect to see on the graph of \(f(x)=x^{2}+4 x+5\)?
Graph \(f(x)=x^{2}+4 x+5\) by using its properties.
Solution :
| Since \(a\) is \(-1\), the parabola opens downward. | |
| To find the equation of the axis of symmetry, use \(x=-\frac{b}{2 a}\). | |
|
The equation of the axis of symmetry is \(x=-2). |
|
| The vertex is on the line \(x=-2\). | |
| Find \(f(x)\) when \(x=-2\). | |
|
The vertex is \((-2,1)\). |
|
| The \(y\)-intercept occurs when \(x=0\). | |
| Find \(f(0)\). | |
| Simplify. | |
| The \(y\)-intercept is \((0,5)\). | |
| The point \((-4,5)\) is two units to the left of the line of symmetry. The point to units to the right of the line of symmetry is \((0,5)\. | |
| Point symmetric to the \(y\)-intercept is \((-4,5)\). | |
| The \(x\)-intercept occurs when \(f(x)=0\). | |
| Find \(f(x)=0\). | |
| Test the discriminant. | |
| Since the value of the discriminant is negative, there is no real solution and so no \(x\)-intercept. | |
| Connect the points to graph the parabola. You may want to choose two more points for greater accuracy. |
Graph \(f(x)=x^{2}-2 x+3\) by using its properties.
- Answer
Graph \(f(x)=-3x^{2}-6 x-4\) by using its properties.
- Answer
Finding the \(y\)-intercept by finding \(f(0)\) is easy, isn't it? Sometimes we need to use the Quadratic Formula to find the \(x\)-intercepts.
Graph \(f(x)=2 x^{2}-4 x-3\) by using its properties.
Solution :
|
Since \(a\) is \(2\), the parabola opens upward. |
|
| To find the equation of the axis of symmetry, use \(x=-\frac{b}{2 a}\). | \(x=-\frac{b}{2 a}\) |
| \(x=-\frac{-4}{2 \cdot 2}\) | |
| \(x=1\) | |
| The equation of the axis of symmetry is \(x=1\). | |
| The vertex is on the line \(x=1\). | \(f(x)=2 x^{2}-4 x-3\) |
| Find \(f(1)\). | |
| \(f(1)=2-4-3\) | |
| \(\f(1)=-5) | |
| The vertex is \((1,-5)\). | |
| The \(y\)-intercept occurs when \(x=0\). | \(f(x)=2 x^{2}-4 x-3\) |
| Find \(f(0)\). | |
| Simplify. | \(f(0)=-3\) |
| The \(y\)-intercept is \((0,-3)\). | |
| The point \((0,-3)\) is one unit to the left of the line of symmetry. | Point symmetric to the \(y\)-intercept is \((2,-3)\) |
| The point one unit to the right of the line of symmetry is \((2,3)\). | |
| The \(x\)-intercept occurs when \(y=0\). | \(f(x)=2 x^{2}-4 x-3\) |
| Find \(f(x)=0\). | |
| Use the Quadratic Formula. | \(x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\) |
| Substitute in the values of \(a,b\) and \(c\). | \(x=\frac{-(-4) \pm \sqrt{(-4)^{2}-4(2)(3)}}{2(2)}\) |
| Simplify. | \(x=\frac{-4 \pm \sqrt{16+24}}{4}\) |
| Simplify inside the radical. | \(x=\frac{4 \pm \sqrt{40}}{4}\) |
| Simplify the radical. | \(x=\frac{4 \pm 2 \sqrt{10}}{4}\) |
| Factor the GCF. | \(x=\frac{2(2 \pm \sqrt{10})}{4}\) |
| Remove common factors. | \(x=\frac{2 \pm \sqrt{10}}{2}\) |
| Write as two equations. | \(x=\frac{2+\sqrt{10}}{2}, \quad x=\frac{2-\sqrt{10}}{2}\) |
| Approximate the values. | \(x \approx 2.5, \quad x \approx-0.6\) |
| The approximate values of the \(x\)-intercepts are \((2.5,0)\) and \((-0.6,0)\). | |
| Graph the parabola using the points found. |
Graph \(f(x)=5 x^{2}+10 x+3\) by using its properties.
- Answer
Graph \(f(x)=-3 x^{2}-6 x+5\) by using its properties.
- Answer
Solve Maximum and Minimum Applications
Knowing that the vertex of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic function. The y -coordinate of the vertex is the minimum value of a parabola that opens upward. It is the maximum value of a parabola that opens downward. See Figure 9.6.124 .
Minimum or Maximum Values of a Quadratic Function
The y -coordinate of the vertex of the graph of a quadratic function is the
- minimum value of the quadratic equation if the parabola opens upward .
- maximum value of the quadratic equation if the parabola opens downward .
Find the minimum or maximum value of the quadratic function \(f(x)=x^{2}+2 x-8\).
Solution :
| \(f(x)=x^{2}+2 x-8\) | |
| Since \(a\) is positive, the parabola opens upward. The quadratic equation has a minimum. | |
| Find the equation of the axis of symmetry. | \(x=-\frac{b}{2 a}\) |
| \(x=-\frac{2}{2 \times 1}\) | |
| \(x=-1\) | |
| The equation of the axis of symmetry is \(x=-1\). | |
| The vertex is on the line \(x=-1\). | \(f(x)=x^{2}+2 x-8\) |
| Find \(f(-1)\). | |
| \(f(-1)=1-2-8\) | |
| \(f(-1)=-9\) | |
| The vertex is \((-1,-9)\). | |
| Since the parabola has a minimum, the \(y\)-coordinate of the vertex is the minimum \(y\)-value of the quadratic equation. The minimum value of the quadratic is \(-9\) and it occurs when \(x=-1\). | |
Show the graph to verify the result.
Find the maximum or minimum value of the quadratic function \(f(x)=x^{2}-8 x+12\).
- Answer
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The minimum value of the quadratic function is \(−4\) and it occurs when \(x=4\).
Find the maximum or minimum value of the quadratic function \(f(x)=-4 x^{2}+16 x-11\).
- Answer
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The maximum value of the quadratic function is \(5\) and it occurs when \(x=2\).
We have used the formula
\(h(t)=-16 t^{2}+v_{0} t+h_{0}\)
to calculate the height in feet, \(h\), of an object shot upwards into the air with initial velocity, \(v_{0}\), after \(t\) seconds .
This formula is a quadratic function, so its graph is a parabola. By solving for the coordinates of the vertex \((t,h)\), we can find how long it will take the object to reach its maximum height. Then we can calculate the maximum height.
The quadratic equation \(h(t)=-16 t^{2}+176 t+4\) models the height of a volleyball hit straight upwards with velocity \(176\) feet per second from a height of \(4\) feet.
- How many seconds will it take the volleyball to reach its maximum height?
- Find the maximum height of the volleyball.
Solution :
\(h(t)=-16 t^{2}+176 t+4\)
Since \(a\) is negative, the parabola opens downward. The quadratic function has a maximum.
a. Find the equation of the axis of symmetry.
\(\begin{array}{l}{t=-\frac{b}{2 a}} \\ {t=-\frac{176}{2(-16)}} \\ {t=5.5}\end{array}\)
The equation of the axis of symmetry is \(t=5.5\).
The vertex is on the line \(t=5.5\).
The maximum occurs when \(t=5.5\) seconds.
b. Find \(h(5.5)\).
\(\begin{array}{l}{h(t)=-16 t^{2}+176 t+4} \\ {h(t)=-16(5.5)^{2}+176(5.5)+4}\end{array}\)
Use a calculator to simplify.
\(h(t)=488\)
The vertex is \((5.5,488)\).
Since the parabola has a maximum, the \(h\)-coordinate of the vertex is the maximum value of the quadratic function.
The maximum value of the quadratic is \(488\) feet and it occurs when \(t=5.5\) seconds.
After \(5.5\) seconds, the volleyball will reach its maximum height of \(488\) feet.
Solve, rounding answers to the nearest tenth.
The quadratic function \(h(t)=-16 t^{2}+128 t+32\) is used to find the height of a stone thrown upward from a height of \(32\) feet at a rate of \(128\) ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height?
- Answer
-
It will take \(4\) seconds for the stone to reach its maximum height of \(288\) feet.
A path of a toy rocket thrown upward from the ground at a rate of \(208\) ft/sec is modeled by the quadratic function of \(h(t)=-16 t^{2}+208 t\). When will the rocket reach its maximum height? What will be the maximum height?
- Answer
-
It will take \(6.5\) seconds for the rocket to reach its maximum height of \(676\) feet.
Key Concepts
-
Parabola Orientation
-
For the graph of the quadratic function \(f(x)=a x^{2}+b x+c\), if
- \(a>0\), the parabola opens upward.
- \(a<0\), the parabola opens downward.
-
For the graph of the quadratic function \(f(x)=a x^{2}+b x+c\), if
-
Axis of Symmetry and Vertex of a Parabola The graph of the function \(f(x)=a x^{2}+b x+c\) is a parabola where:
- the axis of symmetry is the vertical line \(x=-\frac{b}{2 a}\).
- the vertex is a point on the axis of symmetry, so its \(x\)-coordinate is \(-\frac{b}{2 a}\).
- the \(y\)-coordinate of the vertex is found by substituting \(x=-\frac{b}{2 a}\) into the quadratic equation.
-
Find the Intercepts of a Parabola
-
To find the intercepts of a parabola whose function is \(f(x)=a x^{2}+b x+c\):
-
\(y\)-intercept
- Let \(x=0\) and solve for \(f(x)\).
-
\(x\)-intercepts
- Let \(f (x)= 0\) and solve for \(x\).
-
\(y\)-intercept
-
To find the intercepts of a parabola whose function is \(f(x)=a x^{2}+b x+c\):
-
How to graph a quadratic function using properties.
- Determine whether the parabola opens upward or downward.
- Find the equation of the axis of symmetry.
- Find the vertex.
- Find the \(y\)-intercept. Find the point symmetric to the y -intercept across the axis of symmetry.
- Find the \(x\)-intercepts. Find additional points if needed.
- Graph the parabola.
-
Minimum or Maximum Values of a Quadratic Equation
- The \(y\)-coordinate of the vertex of the graph of a quadratic equation is the
- minimum value of the quadratic equation if the parabola opens upward .
- maximum value of the quadratic equation if the parabola opens downward .
Glossary
- quadratic function
- A quadratic function, where \(a, b\), and \(c\) are real numbers and \(a≠0\), is a function of the form \(f(x)=ax^{2}+bx+c\).


13.17: Printing Records
Learning Outcomes
- Preview and print records
Previewing and printing objects in Access follows the same paradigm as most Office applications. Selecting “File” – “Print” opens a dialog box as shown in the example below.
“Quick Print” sends the print request directly to the printer without any other options presented. “Print Preview” displays the print as it will be rendered and printed. “Print” offers all of the usual options from the Office suite such as: Print Range, Copies, Collate, Printer, Properties, Margins, etc.
Contributors and Attributions
CC licensed content, Original
- Printing Records. Authored by : Robert Danielson. Provided by : Lumen Learning. License : CC BY: Attribution


31.0: Introduction
One morning in a suburban Texas emergency department (ED), Eileen, a self-identified 16-year-old female, presented for treatment, clutching her abdomen and complaining of pain. Following an assessment, the ED nurse could not find anything wrong with Eileen. When told she could go home, Eileen said if she went home, she’d “kill myself by taking a whole bottle of extra-strength Tylenol.” Since Eileen could not be discharged if her life was in danger, she was moved to a bed in the hospital’s inpatient mental health unit, where psychiatric nurse Sally Strong was in the middle of her shift. Nurse Strong scrolled through Eileen’s chart to see if she had any previous mental-health visits or a history of depression . Finding none, Nurse Strong noticed that Eileen had recently been to several nearby EDs, each time receiving treatment for sexually transmitted infections (STIs) and various types of substance misuse (marijuana, cocaine, ecstasy, and bath salts). A few months earlier, Nurse Strong’s hospital implemented a protocol that helped nurses identify victims of sex trafficking. Eileen’s repeated ED visits for STIs and substance misuse suggested to Nurse Strong that Eileen might be one of them, and her instincts were right. She sat with Eileen and gently shared that she noticed her medical history, asking, “Is someone making you do things you don’t want to do?”  Eileen did not have to say the exact words for Nurse Strong to know Eileen was being trafficked. Nurse Strong followed the hospital protocol for reporting. She referred Eileen to a safe house and connected her with resources to help her build a new life.
A community crisis can be extremely challenging for the people directly affected. It can include individuals, local families, workers, businesses, industries, and the community. As nurses, listening to the voices of the people impacted by the crisis is essential to build trust, foster resilience, and facilitate recovery. One only needs to watch the global news media to recognize an exponential number of communities in crisis—from both natural and human causes. This ongoing expansion requires a simultaneous evolutionary adaptation in nursing education, practice, research, and policy to ensure that all nurses—from front-line professionals to researchers—have the baseline knowledge, skills, abilities, and autonomy to protect populations at greatest risk and improve the readiness, safety, and support of the nursing workforce.


5.5: Assessment
True/False
Indicate whether the statement is true or false.
____ 1. An attacker has successfully committed a denial-of-service attack against a website, bringing it down for three hours until network engineers could resolve the problem. This is classified as a threat.
____ 2. Vulnerabilities are weaknesses that allow a threat to occur.
____ 3. Attacks require malicious intent, so they are always caused by people who intend to violate security.
____ 4. Lightning is an example of a threat agent.
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 5. Which of the following is not an example of a threat category?
| a. Attacks | c. Natural event |
| b. Buggy software | d. Human error |
____ 6. Which of the following is not a threat to critical infrastructure?
| a. Availability of very sophisticated tools that don’t require much skill to use | c. The rapid development of technology |
| b. The high-profile nature of critical infrastructure systems | d. The interconnected nature of industrial control systems |
____ 7. An attacker that breaks into computers for profit or bragging rights is a/an . . .
| a. Cracker | c. Terrorist |
| b. Insider | d. Hostile country |
Completion
Complete the sentence.
8. The types of attacks and attackers specific to a company is known as the threat ___________.
9. A social engineering attack in which victims are tricked into clicking an emailed link that infects their system with malware or sends their user IDs and passwords to the attacker is known as ____________.
10. A security control that creates a list of authorized applications, preventing unauthorized applications from downloading and installing, is called a/an ___________.
Matching
Match each threat to its definition.
| A. Denial-of-service (DoS) attack | F. SQL injection |
| B. Hijacking | G. Trojan horse |
| C. Ransomware | H. Virus |
| D. Distributed denial-of-service (DDoS) attack | I. SMTP spam engine |
| E. Buffer overflow | J. Worm |
____ 11. An attack in which multiple attackers attempt to flood a device
____ 12. Malware that replicates autonomously
____ 13. A web application attack against a connected database
____ 14. Malicious code attached to a file that, when executed, delivers its payload
____ 15. Malware that encrypts the victims files on their computer until money is sent to the attacker
____ 16. An attack that leverages email protocols to send out messages from the infected device
____ 17. An attack that seizes control of communications, sending the communications to the attacker’s system
____ 18. An attack in which a single attacker overwhelms a system with a flood of traffic in order to make it unavailable
____ 19. An attack that writes data to unexpected areas of memory, causing the device to crash
____ 20. Malware embedded in what appears to be a useful file
For the answers to these questions, email your name, the name of your college or other institution, and your position there to<EMAIL_ADDRESS>. CyberWatch West will email you a copy of the answer key.


4.3: Gray Matters for the Hughes Aircraft Case
Introduction
I. Introduction: The Hughes Aircraft Case involves a group of employees in charge of testing chips for weapons systems. Because of the lengthy testing procedure required by the U.S. Defense Department, Hughes soon fell behind schedule in delivering chips to customers. To get chips out faster, some Hughes middle-level managers began to put pressure on employees to pass chips that had failed tests or to pass them without testing. The scenarios below consist of narratives that stop at the point of decision. Your job is to complete the narrative by making a decision. Alternatives are provided to get the process started, but you may find it necessary to design your own solution. Ethics and feasibility tests help you to evaluate these alternatives and even design new ones more to your liking. This format superficially resembles the Gray Matters exercise used at Boeing Corporation. (More information on the history of Gray Matters can be found by consulting Carolyn Whitbeck, Ethics in Engineering Practice, 1998, 176-182.) This version differs in being more open-ended and more oriented toward giving you the opportunity to practice using ethical theory (which has been encapsulated into ethics tests).
Directions
- Read the following scenarios and the accompanying solutions
- Evaluate the alternatives in terms of the tests described below.
- Choose the one you think best or design your own solution if you believe you can do better.
- Summarize your results by filling in the solution evaluation matrix that appears on the page following the scenario. Notice that the first column repeats the solution alternatives.
- Be prepared to present your matrix to the class. You will also provide the other groups in the class with a copy of your matrix for their ethics portfolios
Bibliographical Note: The six scenarios below were developed by Chuck Huff as Participant Perspectives. They were first published online through the Computing Cases website. (Computing Cases was developed through two National Science Foundation grants, DUE-9972280 and DUE-9980768.) A revised version of these participant perspectives has been published in the anthology, Whistleblowing: Perspectives and Experiences, edited by Reena Raj and published in 2008 by the Icfai University Press, Nagarjuna Hills, Punjagutta, Hyderbad, India. These materials can be found on pages 75-80.
Scenario One: Responding to Organizational Pressure: Frank Saia has worked at Hughes Aircraft for a long time. Now he is faced with the most difficult decisions of his career. He has been having problems in the environmental testing phase of his microchip manufacturing plant; the detailed nature of these tests has caused Hughes to be consistently late in delivering the chips to customers. Because of the time pressure to deliver chips, Saia has been working to make the production of chips more efficient without losing the quality of the product. Chips are manufactured and then tested, and this provides two places where the process can bottle up. Even though you might have a perfectly fine chip on the floor of the plant, it cannot be shipped without testing. And, since there are several thousand other chips waiting to be tested, it can sit in line for a long time. Saia has devised a method that allows testers to put the important chips, the “hot parts,” ahead of the others without disrupting the flow and without losing the chips in the shuffle. He has also added a “gross leak” test that quickly tells if a chip in a sealed container is actually sealed or not. Adding this test early in the testing sequence allows environmental testing to avoid wasting time by quickly eliminating chips that would fail a more fine-grained leak test later in the sequence. Because environmental testing is still falling behind, Saia’s supervisors and Hughes customers are getting angry and have begun to apply pressure. Karl Reismueller, the director of the Division of Microelectronics at Hughes, has given Saia’s telephone number to several customers, whose own production lines were shut down awaiting the parts that Saia has had trouble delivering. His customers are now calling him directly to say “we’re dying out here” for need of parts. Frank Saia has discovered that an employee under his supervision, Donald LaRue, has been skipping tests on the computer chips. Since LaRue began this practice, they have certainly been more on time in their shipments. Besides, both LaRue and Saia know that many of the “hot” parts are actually for systems in the testing phase, rather than for ones that will be put into active use. So testing the chips for long-term durability that go into these systems seems unnecessary. Still, LaRue was caught by Quality Control skipping a test, and now Saia needs to make a decision. Upper management has provided no guidance; they simply told him to “handle it” and to keep the parts on time. He can’t let LaRue continue skipping tests, or at least he shouldn’t let this skipping go unsupervised. LaRue is a good employee, but he doesn’t have the science background to know which tests would do the least damage if they were skipped. He could work with LaRue and help him figure out the best tests to skip so the least harm is done. But getting directly involved in skipping the tests would mean violating company policy and federal law.
Alternatives:
- Do nothing. LaRue has started skipping tests on his own initiative. If any problems arise, then LaRue will have to take responsibility, not Saia, because LaRue was acting independently of and even against Saia’s orders.
- Call LaRue in and tell him to stop skipping tests immediately. Then call the customers and explain that the parts cannot be shipped until the tests are carried out.
- Consult with LaRue and identify nonessential chips or chips that will not be used in systems critical to safety. Skipping tests on these chips will do the least damage.
- Your solution…
Scenario Two: Responding to WrongdoingMargaret Goodearl works in a supervisory position in the environmental testing group at Hughes Aircraft. Her supervisor, Donald LaRue, is also the current supervisor for environmental testing. The group that LaRue and Goodearl together oversee test the chips that Hughes makes in order to determine that they would survive under the drastic environmental conditions they will likely face. Rigorous testing of the chips is the ideal, but some chips (the hot chips) get in line ahead of others. Goodearl has found out that over the last several months, many of these tests are being skipped. The reason: Hughes has fallen behind in the production schedule and Hughes upper management and Hughes customers have been applying pressure to get chip production and testing back on schedule. Moreover, LaRue and others feel that skipping certain tests doesn’t matter, since many of these chips are being used in systems that are in the testing phase, rather than ones that will be put into active use. A few months after Margaret Goodearl started her new position, she was presented with a difficult problem. One of the “girls” (the women and men in Environmental Testing at Hughes), Lisa Lightner, came to her desk crying. She was in tears and trembling because Donald LaRue had forcefully insisted that she pass a chip that she was sure had failed the test she was running. Lightner ran the hermeticity test on the chips. The chips are enclosed in a metal container, and one of the questions is whether the seal to that container leaks. From her test, she is sure that the chip is a “leaker”—the seal is not airtight so that water and corrosion will seep in over time and damage the chip. She has come to Goodearl for advice. Should she do what LaRue wants and pass a chip she knows is a leaker?
Alternatives:
- Goodearl should advise Lightner to go along with LaRue. He is her supervisor. If he orders to pass the chip, then she should do so.
- Goodearl should go to Human Resources with Lightner and file a harassment complaint against LaRue. Skipping tests is clearly illegal and ordering an employee to commit an illegal act is harassment.
- Goodearl and Lightner should blow the whistle. They should go to the U.S. defense department and inform them of the fact that Hughes Aircraft is delivering chips that have either failed tests or have not been tested.
- Your solution…
Scenario 3: Goodearl, Ibarra, and the AMRAAM Incident: Now that Goodearl had few sympathizers among upper management, she increasingly turned to Ruth Ibarra in Quality assurance for support in her concerns about test skipping and the falsification of paperwork. One day, Goodearl noticed that some AMRAAM chips with leak stickers were left on her project desk in the environmental testing area. The leak stickers meant that the seal on the chips' supposedly airtight enclosure had failed a test to see if they leaked. AMRAAM meant that the chips were destined to be a part of an Advanced Medium Range Air-to-Air Missile. Goodearl knew that these parts could not be retested and needed to be simply thrown away. So why was someone keeping them? She also knew that these were officially "hot parts" and that the company was behind schedule in shipping these parts. After consulting with Ruth Ibarra, the two of them decided to do some sleuthing. They took the chips and their lot travelers to a photocopy machine and made copies of the travelers with "failed" noted on the leak test. They then replaced the chips and their travelers on the desk. Later that day, as Don LaRue passed the desk, Goodearl asked Don LaRue if he knew anything about the chips. "None of your business," he replied. The chips disappeared, and later the travelers showed up in company files with the "failed" altered to "passed." So, Goodearl and Ibarra had clear evidence (in their photocopy of the "failed" on the traveler) that someone was passing off failed chips to their customers. And these were important chips, part of the guidance system of an air-to-air missile.
Alternatives: Since they have clear evidence, Goodearl and Ibarra should blow the whistle. Evaluate each of the following ways in which they could blow the whistle
- Blow the whistle to Hughes’ Board of Directors. In this way they can stop the test skipping but will also be able to keep the whole affair “in house.”
- Blow the whistle to the local news media. In this way they will shame Hughes into compliance with the testing requirements.
- Take the evidence to the U.S. Department of Defense, since they are the client and are being negatively impacted by Hughes’ illegal actions.
- Some other mode of blowing the whistle….
| Alternatives/Tests | Reversibility/Rights Test | Harm/Benefits Test | Virtue/Value Test (Also Publicity) | Global Feasibility Test (Implementation Obstacles) |
| Alternative One (Worst Alternative) | Evaluate Alt 1 using reversibility/rights test | |||
| Alternative Two (Best among those given) | Weigh harms against benefits for alt 2 | |||
| Alternative Three | What values/disvalues are realized in alt 3? | |||
| Your Solution | What obstacles could hinder implementation of solution? |
Ethics Tests: Set Up and Pitfalls
III. Solution Evaluation Tests
- REVERSIBILITY: Would I think this is a good choice if I were among those affected by it?
- PUBILICITY: Would I want to be publicly associated with this action through, say, its publication in the newspaper?
- HARM/BENEFICENCE: Does this action do less harm than any of the available alternatives?
- FEASIBILITY: Can this solution be implemented given time, technical, economic, legal, and political constraints?
Harm Test Set-Up
- Identify the agent (=the person who will perform the action). Describe the action (=what the agent is about to do).
- Identify the stakeholders (individuals who have a vital interest at risk) and their stakes.
- Identify, sort out, and weight the expected results or consequences.
Harm Test Pitfalls
- Paralysis of Action--considering too many consequences
- Incomplete analysis--considering too few results
- Failure to weigh harms against benefits
- Failure to compare different alternatives
- Justice failures--ignoring the fairness of the distribution of harms and benefits
Reversibility Test Set-Up
- Identify the agent
- Describe the action
- Identify the stakeholders and their stakes
- Use the stakeholder analysis to select the relations to be reversed.
- Reverse roles between the agent (you) and each stakeholder: put them in your place (as the agent) and yourself in their place (as the target of the action
- If you were in their place, would you still find the action acceptable?
Reversibility Pitfalls
- Leaving out a key stakeholder relation
- Failing to recognize and address conflicts between stakeholders and their conflicting stakes
- Confusing treating others with respect with capitulating to their demands (Reversing with Hitler)
- Failing to reach closure, i.e., an overall global reversal assessment that takes into account all the stakeholders the agent has reversed with
Public Identification Set-Up
- Set up the analysis by identifying the agent, describing the action under consideration, and listing the key values or virtues at play in the situation
- Associate the action with the agent
- Identify what the action says about the agent as a person. Does it reveal him or her as someone associated with a virtue/value or a vice?
Public Identification Pitfalls
- Action is not associated with the agent. The most common pitfall is failure to associate the agent and the action. The action may have bad consequences and it may treat individuals with disrespect but these points are not as important in the context of this test as what they imply about the agent as a person who deliberately performs such an action
- Failure to specify the moral quality, virtue, or value of the action that is imputed to the agent in the test. To say, for example, that willfully harming the public is bad fails to zero in on precisely what moral quality this attributes to the agent. Does it render him or her unjust, irresponsible, corrupt, dishonest, or unreasonable?
This timeline is taken from the Computing Cases website developed and maintained by Dr. Charles Huff at St. Olaf College. Computing Cases is funded by the National Science Foundation, NSF DUE-9972280 and DUE 9980768.
| 1979 | Ruth Ibarra begins working for Hughes Aircraft company's Microelectronic Circuit Division (Hughes MCD) in Newport Beach, CA |
| 1981 | Margaret Goodearl begins working for Hughes MCD as a supervisor for assembly on the hybrid production floor and as a supervisor in the hybrid engineering lab |
| 1984 | Ibarra becomes supervisor for hybrid quality assurance |
| 1985 | Goodearl asks Ibarra to look at errors in paperwork, Ibarra brings errors to the attention of her supervisors and was told to keep quiet. This begins time period where Goodearl/Ibarra become aware of problems in hybrid chip testing and paperwork. |
| 1986 | Goodearl becomes supervisor for seals processing in the environmental testing area. |
| 1986 | False Claims Act (31 U.S. C 3729-3733) becomes False Claims Reform Act of 1986 making it stronger and easier to apply. |
| Oct. 1986 | Goodearl/Ibarra report problems ot Hughes management, and, after the problems were not fixed, Goodearl/Ibarra reported the allegations of faulty testing to the United States Department of Defense. |
| Jan 9, 1987 | Earliest date that Hughes may have stopped neglecting environmental screening tests. |
| 1988 | Ibarra leaves Hughes feeling that her job had been stripped of all real responsibility. |
| March 1989 | Goodearl is laid off from Hughes. |
| 1995 | Goodearl and her husband are divorced. |
| 1990-1996 | United States of America, ex rel. Taxpayers Against Fraud, Ruth Aldred (was Ibarra), and Margaret Goodearl v. Hughes Aircraft Company, Inc. |
| 1990 | Goodearl files wrongful discharge suit against Hughes and a number of individual managers, which was eventually dropped in favor of the civil suit. |
| May 29, 1990 | Thinking the government investigation was taking too much time, Goodearl/Aldred file civil suit against Hughes under False Claims Reform Act of 1986 with the help of Taxpayers Against Fraud and Washington law firm Phillips and Cohen. |
| December 1992 | Under provisions of the FCA, the U.S. Department of Justice Civil Division takes over the civil case. |
| Sep. 10, 1996 | Hughes found guilty in civil trial. Pays U.S. Government 4,050,00 dollars and each relator 891,000 dollars plus a separate payment of 450,000 dollars to cover attorney's fees, costs, and expenses. |
| 1991-1993 | United States of America v. Hughes Aircraft Co., and Donald LaRue |
| December 13, 1991 | After a lengthy investigation, the U.S. Department of Defense charges Hughes and Donald A. LaRue with a 51-count indictment accusing it of falsifying tests of microelectronic circuits (criminal suit). |
| June 15, 1992 | Hughes found guilty of conspiring to defraud the U.S. Government in crminal case, co-defendent LaRUE acquitted following 4-week trial. Goodearl/Aldred called as witnesses in trial. Hughes appeals. |
| Oct. 29, 1992 | Hughes fined 3.5 million in criminal trial decision. |
| December 2, 1993 | Appellate court upholds 1992 criminal conviction and sentence. Hughes appeals. |
Hughes Case Socio Technical System
| Hardware/Software | Physical Surroundings | People, Roles, Structures | Procedures | Laws and Regulations | Data and Data Structures | |
| Description | Hybrid Chips (circuitry hermetically sealed in metal or ceramic packages in inert gas atmosphere | Battle conditions under which chips might be used | Hughes Microelectric Circuit Division | Chip Testing: Temperature Cycle, Constant Acceleration, Mechanical Shock, Hermeticity (Fine and Gross Leak), P.I.N.D. | Legally Mandated Tests | Lot Travelors to document chips |
| Analogue to Digital Conversion Chips | E-1000 at Hughes (Clean Room) | Department of Defense (Office of Inspector General) | Hughes Human Resources Procedures for Complaints | Whistle Blower Protection Legislation | ||
| Radar and Missile Guidance Systems | Hughes Quality Control | Dissenting Professional Opinions | Qui Tam Lawsuit, Civil Suit, Criminal Suit | |||
| Individuals: Reismueller, Temple, Saia, LaRue, Goodearl, Ibarra/Aldren |
Responsible Dissent
Sources:
- Computing Cases is the primary source for the material below on responsible dissent. It is based on the materials for responsibly carrying out dissent and disagreement that was formerly posted at the IEEE website. The IEEE has since taken this material down.
- The Online Ethics Center has also posted the IEEE material on responsible dissent. The origin of this material as well as a thorough discussion of its content can be found in Carolyn Whitbek, Ethics in Engineering Practice and Research: 2nd Edition, Cambridge, UK: Cambridge University Press, 2011. Chapter 7, "Workplace Rights and Responsibilities, pp. 227-269.
- Much of this material (IEEE Guidelines and a discussion of Dissenting Professional Opinion Guidelines) can be found in Chapter 7 ("Averting the Conflict at the Source")in the following: Stephen H., Unger, Controlling Technology: Ethics and the Responsible Engineer: 2nd Edition, New York: John Wiley and Sons, INC.
Generic Forms of Dissent:
- Gather more information
- Nolo Contendere (Don't fight it. Go along.)
- Oppose diplomatically: Offer your supervisor alternatives to the wrong he or she has ordered.
- Oppose by confronting: Threaten to go over your supervisor's head or threaten to blow the whistle
- Distance yourself: Ask to be transferred to another section to avoid being implicated in the wrongdoing
- Exit: Quit and do nothing or quit and blow the whistle
- Document your position: If your company has a Dissenting Professional Opinion process. If it doesn't, work to have one implemented. By establishing your opposition, you distance yourself morally from the wrong you have documented
- Which one is right? Use your tests. Which does the best job of satisfying the three ethics tests of reversibility, harm, and publicity? Which does the best job with the ADEM values: justice, responsibility, respect, trust, and integrity?
Introduction to Circumstances of Compromise: The following presents the circumstances of compromise as laid down by Martin Benjamin in Splitting the Difference. (See below for complete reference.) Benjamin provides five conditions that indicate when a compromise may be necessary. But he also argues that integrity helps draw a line beyond which compromise must not go. One should not sacrifice basic beliefs that constitute one's personal identity or self-system. A good example of using integrity to draw the line on compromise can be found in the characterization of Thomas More in Robert Bolt's A Man for All Seasons. (See also the movie of the same name.) In the preface to the play, Bolt explicitly presents what follows as an exercise in articulating and testing integrity.
Circumstances of Compromise:
- Under these conditions, it may be necessary to "split the difference."
- Factual uncertainty
- Moral complexity
- Continuing Cooperative Relationship
- Decision cannot be deferred
- Scarcity of Resources
More on Moral Complexity: Martin Benjamin in Splitting the Difference quotes John Rawls on moral complexity: “Diversity naturally arises from our limited powers and distinct perspectives; it is unrealistic to suppose that all our differences are rooted in ignorance and perversity, or else in the rivalries that result from scarcity….Deep and unresolvable differences on matters of fundamental significance…[must be acknowledged] as a permanent condition of human life.”
Application of Circumstances of Compromise:
- Factual Uncertainty: Where are the chips under consideration going? If they go to an essential system in an operative technology, then their malfunctioning could lead to loss of life. If they go to a non-essential system (like a prototype being tested) then maybe the testing process can be streamlined. This may require compromise between Hughes management, chip-testing team, and customers.
- Moral Complexity: How should an employee like LaRue weigh his loyalty to supervisors and company and his obligation to the public and client? Setting aside his harassment of Gooderal, is Saia's position (or at least a part of it) morally defensible?
- Continuing cooperative relationship: How important should it be to Gooderal that she needs to sustain her relationship with her supervisor, LaRue, for the long term? How important is it that Hughes managers respond to difficult messages rather than attempt to "shoot the messenger." (Again, thinking in terms of continuing cooperative relationship?)
- Decision cannot be deferred: Why is it impossible to defer the decision on whether to respond to test skipping? This case poses several difficult constraints. How many of these can be "pushed back" through negotiation? Could Saia use his newly found accessibility to customers to negotiate with them an extension on the delivery deadlines?
- Scarcity of resources: How are the resources of time, personnel, and money scarce in this case? Is there any way to push back these constraints by negotiating more time (extending deadlines for delivering chips), personnel (bringing in additional people to test chips), and resources (developing better tools to test chips more quickly). Could, for example, it be possible to transfer Hughes employees from other areas to help out, temporarily, on chip testing?
Ethical Dissent:
- Establish a clear technical foundation.
- Keep your arguments on a high professional plane, as impersonal and objective as possible, avoiding extraneous issues and emotional outbursts.
- Try to catch problems early, and keep the argument at the lowest managerial level possible.
- Before going out on a limb, make sure that the issue is sufficiently important.
- Use (and help establish) organizational dispute resolution mechanisms.
- Keep records and collect paper.
- These items originate with the IEEE which has dropped them from their website. They can be accessed through the link above with the Online Ethics Center; the list there is more complete. The above is quoted from the Computing Cases website: http://computingcases.org/case_mater...l_dissent.html .
Before Going Public:
- Make sure of your motivation.
- Count your costs.
- Obtain all the necessary background materials and evidence.
- Organize to protect your own interests.
- Choose the right avenue for your disclosure.
- Make your disclosure in the right spirit.
- These items come from the IEEE (see onlineethics link) and from the manuscript of Good Computing by Chuck Huff, William Frey, and Jose Cruz.
Places to Go:
- Government Agencies
- Judicial Systems
- Legislators
- Advocacy Groups
- News Media
- In Puerto Rico, laws 14 and 426 have been passed to protect those who would blow the whistle on government corruption. The Oficina de Etica Gubernamental de Puerto Rico has a whistle blower's hotline. See link above.
When to Blow the Whistle:
- Serious and Considerable Harm
- Notification of immediate supervisor.
- Exhaustion of internal channels of communication/appeal.
- Documented Evidence.
- Likelihood of successful resolution.
- When the first three conditions are satisfied, whistle-blowing is morally permissible. (You may do it but you are not required or obligated to do it.) This is because you have brought your concerns before decision-makers, given them a chance to respond, and, in the face of their unwillingness to do so, still find the issue of great importance.
- When all five conditions are satisfied, then whistle-blowing becomes morally obligatory. In this case, you have a moral duty to blow the whistle. Here, your duty is grounded in your responsibility to inform those who are likely to be harmed by the wrongdoing.
References:
- Richard T. De George, "Ethical Responsibilities of Engineers in Large Organizations: The Pinto Case," in Ethical Issues in Engineering, ed. Deborah G. Johnson (1991) New Jersey: Prentice-Hall: 175-186.
- Carolyn Whitbeck (1998) Ethics in Engineering Practice and Research. U.K. Cambridge University Press: 55-72 and 176-181.
- Charles Harris, Michael Pritchard and Michael Rabins (2005) Engineering Ethics: Concepts and Cases, 3rd Ed. Belmont, CA: Thomson/Wadsworth: 203-206.
Hughes Dramatic Rehearsals
A note on dramatic rehearsals:
- The notion of dramatic rehearsal comes from John Dewey's Human Nature and Moral Conduct. An agent works through a solution alternative in the imagination before executing it in the real world. The dramatic rehearsal tests the idea in a mental laboratory created by the moral imagination. Steven Fesmire in his book, John Dewey and Moral Imagination: Pragmatism in Ethics (Indiana University Press, 2003), provides a comprehensive interpretation of Dewey's suggestive idea.
- The scenarios portrayed below reflect events in the case but some changes have been made to create six focused decision points. For a more accurate portrayal of the case events, see Computing Cases (computingcases.org)
Decision Point One
- Frank Saia has worked at Hughes Aircraft for a long time. Now he is faced with the most difficult decisions of his career. He has been having problems in the environmental testing phase of his microchip manufacturing plant; the detailed nature of these tests has caused Hughes to be consistently late in delivering the chips to customers.
- Because of the time pressure to deliver chips, Saia has been working to make the production of chips more efficient without losing the quality of the product. Chips are manufactured and then tested, and this provides two places where the process can bottle up. Even though you might have a perfectly fine chip on the floor of the plant, it cannot be shipped without testing. And, since there are several thousand other chips waiting to be tested, it can sit in line for a long time. Saia has devised a method that allows testers to put the important chips, the “hot parts,” ahead of the others without disrupting the flow and without losing the chips in the shuffle. He has also added a “gross leak” test that quickly tells if a chip in a sealed container is actually sealed or not. Adding this test early in the testing sequence allows environmental testing to avoid wasting time by quickly eliminating chips that would fail a more fine-grained leak test later in the sequence.
- Because environmental testing is still falling behind, Saia’s supervisors and Hughes customers are getting angry and have begun to apply pressure. Karl Reismueller, the director of the Division of Microelectronics at Hughes, has given Saia’s telephone number to several customers, whose own production lines were shut down awaiting the parts that Saia has had trouble delivering. His customers are now calling him directly to say “we’re dying out here” for need of parts.
Dialogue for Decision Point One
- Construct a dialogue in which Saia responds to the pressure from his supervisor, Karl Reismueller
- Be sure to address the customer complaints
Decision Point Two
- Frank Saia has discovered that an employee under his supervision, Donald LaRue, has been skipping tests on the computer chips. Since LaRue began this practice, they have certainly been more on time in their shipments. Besides, both LaRue and Saia know that many of the “hot” parts are actually for systems in the testing phase, rather than for ones that will be put into active use. So testing the chips for long-term durability that go into these systems seems unnecessary. Still, LaRue was caught by Quality Control skipping a test, and now Saia needs to make a decision. Upper management has provided no guidance; they simply told him to “handle it” and to keep the parts on time.
- He can’t let LaRue continue skipping tests, or at least he shouldn’t let this skipping go unsupervised. LaRue is a good employee, but he doesn’t have the science background to know which tests would do the least damage if they were skipped. He could work with LaRue and help him figure out the best tests to skip so the least harm is done. But getting directly involved in skipping the tests would mean violating company policy and federal law.
Dialogue
- Construct a dialogue in which Saia confronts LaRue about skipping the tests
- Address the following issues:
- Should Saia work with LaRue to identify tests that are not necessary and then have LaRue skip these?
- How should Saia and LaRue deal with the concerns that Quality Control has expressed about skipping the tests?
Decision Point Three
- Margaret Goodearl works in a supervisory position in the environmental testing group at Hughes Aircraft. Her supervisor, Donald LaRue, is also the current supervisor for environmental testing. The group that LaRue and Goodearl together oversee test the chips that Hughes makes in order to determine that they would survive under the drastic environmental conditions they will likely face. Rigorous testing of the chips is the ideal, but some chips (the hot chips) get in line ahead of others. Goodearl has found out that over the last several months, many of these tests are being skipped. The reason: Hughes has fallen behind in the production schedule and Hughes upper management and Hughes customers have been applying pressure to get chip production and testing back on schedule. Moreover, LaRue and others feel that skipping certain tests doesn’t matter, since many of these chips are being used in systems that are in the testing phase, rather than ones that will be put into active use.
- A few months after Margaret Goodearl started her new position, she was presented with a difficult problem. One of the “girls” (the women and men in Environmental Testing at Hughes), Lisa Lightner, came to her desk crying. She was in tears and trembling because Donald LaRue had forcefully insisted that she pass a chip that she was sure had failed the test she was running.
- Lightner ran the hermeticity test on the chips. The chips are enclosed in a metal container, and one of the questions is whether the seal to that container leaks. From her test, she is sure that the chip is a “leaker”—the seal is not airtight so that water and corrosion will seep in over time and damage the chip. She has come to Goodearl for advice. Should she do what LaRue wants and pass a chip she knows is a leaker?
Dialogue
- Construct a dialogue that acts out Goodearl’s response to her knowledge that LaRue is regularly skipping tests
- Should Goodearl first talk directly to LaRue? What if he responds defensively?
- Should Goodearl go over LaRue’s head and discuss his skipping the tests with one of his supervisors? To whom should she go? How could she prepare for possible retaliation by LaRue? What should she know before doing this?
- If LaRue or another supervisor should fail to respond to the test skipping, should Goodearl continue responsible dissent or drop the issue (=nolo contendere)
- Could Goodearl not contend the issue but distance herself? (What if Hughes has no DPO procedure?)
Decision Point Four
Ruth Ibarra (from Quality Assurance) has seen Shirley Reddick resealing chips without the authorization stamp. Ibarra has asked Goodearl to find out what’s going on. When Goodearl asks LaRue, he replies, “None of your damn business.” Shortly after this, Gooderal receives a phone call from Jim Temple, one of her superiors, telling her to come to his office. Temple informs Goodearl in no uncertain terms that she needs to back down. “You are doing it again. You are not part of the team, running to Quality with every little problem.” When Goodearl insisted she did not “run to Quality” but Quality came to her, Temple replies, “Shape up and be part of the team if you want your job.”
Dialogue
- Construct a dialogue in which Gooderal responds to this latest test skipping issue
- Consider these issues in constructing your dialogue:
- Goodearl had already confronted LaRue about test skipping when Lisa Lightner came to her. After failing to get results, she had decided to drop the issue
- How should Goodearl respond to Temple? Should she continue pushing responsible dissent or give way to Temple’s threats?
Decision Point Five
- After her conversation with Temple, Goodearl goes to the Personnel Department to inquire into filing a harassment complaint against her supervisors at Hughes
- After her discussion she sees the personnel official leave his office and turn toward Frank Saia’s office, one of Goodearl’s supervisors.
- Goodearl is then called to Saia’s office.  He also asks her where she gets off filing a harassment charge against him.
Dialogue
- Construct a dialogue in which Gooderal reacts to Saia both during Saia’s outburst and after it.
- Is Saia harassing Gooderal? (How do we define “harassing” in this context?)
- How should Goodearl respond given that Saia’s latest outburst was caused by the personnel official reporting to him the confidential meeting he had with Goodearl?
- What are Goodearl’s options at this point? Are any of the strategies for responsible dissent we have studied so far relevant or of use?
Decision Point Six
- Margaret Goodearl and Ruth Ibarra have made several attempts to get their supervisors to respond to the problem of skipping the environmental tests. The general response has been to shoot the messenger rather than respond to the message. Both Goodearl and Ibarra have been branded trouble makers and told to mind their own business. They have been threatened with dismissal if they persist.
- So they have decided to blow the whistle, having exhausted all the other options. They initiated contact with officials in the U.S. government’s Office of the Inspector General. These officials are interested but have told Goodearl and Ibarra that they need to document their case.
- One day they find two hybrids (chips that combine two different kinds of semiconductor devices on a common substrate) on LaRue’s desk. These chips which are destined for an air-to-air missile have failed the leak test. It is obvious that LaRue plans on passing them without further testing during the evening shift after Goodearl has gone home. Goodearl and Ibarra discuss whether this presents a good opportunity to document their case for the Office of the Inspector General.
Dialogue
- Construct an imaginary conversation between Goodearl and Ibarra where they discuss different strategies for documenting their concerns to the Office of the Inspector General?
- Have them consider the following:
- By looking for documented evidence against their employer, have Goodearl and Ibarra violated their duties of trust and confidentiality?
- Some argue that before blowing the whistle, an employee should exhaust internal channels. Have Goodearl and Ibarra discuss whether they can do anything more inside Hughes before taking evidence outside
Questions on Dramatic Reflections
Directions: After you have acted out your decision point in the Hughes case, you and your group have two further activities. First, you will answer the questions below to help you reflect generally on the nature of dramatic rehearsals and specifically on recent dramatization. These five questions (outlined in detail just below) ask you to discuss your dramatic form, the form of responsible dissent you used, how the action you dramatized fared with the three ethics tests, the value and interest conflicts you dealt with, and the constraints that bordered your decision point. Second, you will provide a storyboard that summarizes the drama you acted out before class. This is also detailed just below.
As was said above, John Dewey suggested the idea that underlies these dramatic rehearsals or What-if dramas. As Dewey puts it "[d]eliberation is actually an imaginative rehearsal of various courses of conduct. We give way, in our mind, to some impulse; we try, in our mind, some plan. Following its career through various steps, we find ourselves in imagination in the presence of the consequences that would follow: and as we then like and approve, or dislike and disapprove, these consequences, we find the original impulse or plan good or bad. Deliberation [becomes] dramatic and active…." (Dewey, 1960, p. 135) Think of your "dramatic rehearsal" as an experiment carried out in your imagination. The hypothesis is the alternative course of action decided upon by your group that forms the basis of your "What-if" rehearsal.  What are its consequences? How are these distributed? How would each of the stakeholders in your case view the action? How does this fit in with your conception of a moral or professional career? Your imagination is the laboratory in which you test the action your group devises as a hypothesis.
This quote will also help you understand the concept of dramatic rehearsal. It comes from John Gardner, a famous novelist, and Mark Johnson, a theorist in moral imagination. John Gardner has argued that fiction is a laboratory in which we can explore in imagination the probable implications of people’s character and choices. He describes what he calls “moral fiction” as a “philosophical method” in which art “controls the argument and gives it its rigor, forces the writer to intense yet dispassionate and unprejudiced watchfulness, drives him—in ways abstract logic cannot match—to unexpected discoveries and, frequently, a change of mind.” (Johnson, p. 197; Gardner, p. 108).
There are different versions of what form dramatic form takes. In general there is plot, character, agon (struggle or confronting obstacles), resolution, and closure. A drama is a narrative, an unfolding of related events in time. One event arises to give way to another and so forth. Dramas can be driven by the ends of the characters and the unfolding can be the realization or frustration of these activities. Dramatic rehearsals take isolated actions and restore them to this context of dramatic form.
1. What is the dramatic form taken on by your enactment?
- Perhaps your drama is a comedy. Many groups have chosen this form but have found it hard to explain why. How does comedy help your group to get its message across? What is its message?
- Some groups have approached this rehearsal as a tragedy where the good intentions and goals of those involved all turned out bad. In many ways, this is how the case played out in reality. So, if your group chose tragedy, then it is important to state why there were no viable alternatives to the choices actually made by the participants in the Hughes case. What constraints prevented the agents from achieving their ends? (Look for more than just bad people here.)
- Some groups decided to frame their rehearsal as a documentary Here a narrator describes and frames the activities carried out by the different participants offering commentary and analysis.
- Continuing with the documentary line, some groups have presented their drama as proceedings in a trial where a judge presides over attorneys presenting the arguments from both sides. This approach has the advantage of laying out the different perspectives but when the judge reaches a decision, it takes on the risk of oversimplifying the case by making one side completely right and the other completely wrong. The "winner takes all" interpretation of a court trial (guilty-innocent) often leaves out moral complexity.
- Some groups convert their dramas into Quixotic ventures where they "tilt at windmills." Here they try to present scenarios where idealistic participants strive to realize their values over difficult, constraining and harsh realities. The advantage of this approach is that it does not compromise on values and ideals. The disadvantage is that it may underestimate elements in the real world that oppose acting on the ideal. Realizing the "intermediate possible" may be the best route here.
- Some groups approach their dramatizations as cautionary tales where they act out the harsh consequences that attend immoral, greedy, selfish, or corruption action. Here the world is constrained by justice. Those who hubristically try to exceed these constraints are punished for their transgressions. Cautionary tales are more moralistic than tragedies but, at some point, converge on this other dramatic form.
- You are, of course, encouraged to go beyond this list by inventing your own dramatic form or combining those listed above to produce a new, synthetic form. The point here is that dramatic forms both filter and structure elements of this complicated case. I am asking that you be deliberate and thoughtful about how you work your way through the Hughes case. What did your dramatic form leave out? How did it structure the drama differently than other forms? Why did you choose the form you chose?
2. Your dramatic rehearsal also should test the three forms of responsible dissent we studied this semester.
- Generic Forms of Dissent. Did your rehearsal test any of the generic forms of dissent such as gather more information, nolo contendere, oppose diplomatically, oppose confrontationally, distance yourself, or exit?
- Moral Compromise. Did your rehearsal deploy any of the strategies of moral compromise? For example, referring to the Ethics of Teamwork, did it deploy bridging, logrolling, expanding the pie, or non-specific compensation? Were you able to get things moving by negotiating interests rather than person-based positions? What were the circumstances that elicited compromise? For example, does the Hughes case display any "moral complexity?"
- Blowing the Whistle. If your drama followed the case and advocated blowing the whistle, provide a justification using the class framework. For example, argue that whistleblowing was permissible or that it was obligatory. To whom do your recommend blowing the whistle given the problems Ibarra and Goodearl had with the Inspector General's office? How would you recommend they go about gathering documented evidence? What should they do before blowing the whistle? In short, do more, both in your drama and in this reflection, than just advocate the action. Describes the means, complexities, and circumstances surrounding blowing the whistle on Hughes.
3. Outline your ethics experiment by examining the action you advocate using the three ethics tests.
- Reversibility. How does your action look when you reverse with the key stakeholders? Project into their shoes avoiding the extremes of too much identification and too little identification.
- Harm. What harms have you envisioned through your dramatic rehearsal? Are these harms less quantitatively and qualitatively than the action actually taken in the case?
- Publicity. Finally, project the action taken in your rehearsal into the career of a moral professional. Is it consistent with this career or does it embrace (or neglect) values out of place in such a career. In other words, carry out the publicity test by associating the values embedded in the action you portray with the character of a good or moral agent carrying out a moral, professional career.
4. Value and interest conflicts in your drama.
- All these decision points involve some kind of conflict. How did you characterize this conflict in your dramatization? Pose your conflict in terms of values. How did your drama "resolve" this value conflict?
5. Recognizing and dealing with the constraints you found in your decision point.These drama/decision points had different kinds and degrees of constraints. Early decision points have fewer constraints than later because the earlier decisions both condition and constrain those that follow. Here is another issue you may need to address. Your feasibility test from the "Three Frameworks" module outlines three kinds of constraints: resource, interest (social or personality), and technical. Did any of these apply? Outline these and other constraints and describe how they were dealt with in your drama.
Story Boards
Suggestions for Story Boards
- Divide your dramatization into four to six frmaes. Now draw a picture in each frame, one that captures a key moment of your dramatization.
- Check for continuity. Each frame should present elements that show how it emerges from the previous frame and how it transitions into the subsequent frame. The first frame should help the reader find the context in which your drama takes place. The last frame should provide as much closure as your drama permits.
 But while acting through your drama, you received feedback from the class and, perhaps, began to rethink things. So feel free to make changes in your storyboard to reflect your deeper understanding of your decision point. If you make changes in your storyboard, discuss this in your dramatic reflections. Explain why you decided to change things.
Hughes Case Media Files
Hughes Case and Dialogue Points: Hughes_V3a.pptx , Responsible Dissent.pptx
What If Dramatic Rehearsals: Hughes_Drama_V2.pptx
Debating Topics for ADMI 4016, Spring 2011: Reflections on debate.pptx
Jeopardy: Responsible Dissent: Jeopardy6.pptx
Jeopardy for codes of ethics: Jeo_Codes.pptx Hughes 2014.docx
Bibliography
- Martin Benjamin. (1990). Splitting the Difference: Compromise and Integrity in Ethics and Politics. Lawrence, KS: University of Kansas Press.
- Chuck Huff and William Frey. "The Hughes Whistleblowing Case." In Reena Raj (Ed.) Whistleblowing: Perspectives and Experiences, 75-80. 2008, Hyderabad India: Icfai University Press.
- Charles Harris, Michael Pritchard, Michael Rabins. "Engineers as Employees," in Engineering Ethics: Concepts and Cases, 2nd Edition. Wadsworth Thompson Learning, 2000. Section 8.8 of Chapter 8 discusses DeGeorge's criteria for whistle-blowing.
- Richard T. DeGeorge. "Ethical Responsibilities of Engineers in Large Organizations," in Business and Professional Ethics Journal, Vol 1, no. 1: 1-14.
- Stephen H., Unger, Controlling Technology: Ethics and the Responsible Engineer: 2nd Edition, New York: John Wiley and Sons, INC, 1994.
- Richard T. De George, "Ethical Responsibilities of Engineers in Large Organizations: The Pinto Case," in Ethical Issues in Engineering, ed. Deborah G. Johnson (1991) New Jersey: Prentice-Hall: 175-186.
- Carolyn Whitbeck (1998) Ethics in Engineering Practice and Research. U.K. Cambridge University Press: 55-72 and 176-181. See also 2nd edition (2011) Chapter 7.
- Charles Harris, Michael Pritchard and Michael Rabins (2005) Engineering Ethics: Concepts and Cases, 3rd Ed. Belmont, CA: Thomson/Wadsworth: 203-206.
- Gardner, J. (1978). On Moral Fiction. New York: Basic Books.
- Johnson, M. (1994). Moral Imagination: Implications of Cognitive Science for Ethics. Chicago, IL: University of Chicago Press.


2.2: Beware of Patterns!
The “Look for Patterns” strategy can be particularly appealing, but you have to be careful! Do not forget the “and Explain” part of the strategy. Not all patterns are obvious, and not all of them will continue.
Start with a circle.
If I put two dots on the circle and connect them, the line divides the circle into two pieces.
If I put three dots on the circle and connect each pair of dots, the lines divides the circle into four pieces.
Suppose you put one hundred dots on a circle and connect each pair of dots, meaning every dot is connected to 99 other dots. How many pieces will you get? Lines may cross each other, but assume the points are chosen so that three or more lines never meet at a single point.
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What strategies did you try? What did you figure out? What questions do you still have?
The natural way to work on this problem is to use smaller numbers of dots and look for a pattern, right? If you have not already, try it. How many pieces when you have four dots? Five dots? How would you describe the pattern?
Now try six dots.  Then carefully count up how many pieces you get. It is probably a good idea to work with a partner so you can check each other’s work. Make sure you count every piece once and do not count any piece twice. How can you be sure that you do that?
Were you surprised? For the first several steps, it seems to be the case that when you add a dot you double the number of pieces. But that would mean that for six dots, you should get 32 pieces, and you only get 30 or 31, depending on how the dots are arranged. No matter what you do, you cannot get 32 pieces. The pattern simply does not hold up.
Mathematicians love looking for patterns and finding them. We get excited by patterns. But we are also very skeptical of patterns! If we cannot explain why a pattern would occur, then we are not willing to just believe it.
For example, if my number pattern starts out: 2, 4, 8, ... I can find lots of ways to continue the pattern, each of which makes sense in some contexts. Here are some possibilities:
- 2, 4, 8, 2, 4, 8, 2, 4, 8, 2, 4, 8, ...
This is a a repeating pattern, cycling through the numbers 2, 4, 8 and then starting over with 2.
- 2, 4, 8, 32, 256, 8192, ...
To get the next number, multiply the previous two numbers together.
- 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
- 2, 4, 8, 14, 22, 32, 44, 58, 74 ...
- For the last two patterns above, describe in words how the number sequence is being created.
- Find at least two other ways to continue the sequence 2, 4, 8, . . . that looks different from all the ones you have seen so far. Write your rule in words, and write the next five terms of the number sequence.
So how can you be sure your pattern fits the problem? You have to tie them together! Remember the “Squares on a Chess Board” problem? You might have noticed a pattern like this one:
If the chess board has 5 squares on a side, then there are
- 5 × 5 = 25 squares of size 1 × 1.
- 4 × 4 = 16 squares of size 2 × 2.
- 3 × 3 = 9 squares of side 3 × 3.
- 2 × 2 = 4 squares of size 4 × 4.
- 1 × 1 = 1 squares of size 5 × 5.
So there are a total of
\[1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} = 55 \nonumber \]
squares on a 5 × 5 chess board. You can probably guess how to continue the pattern to any size board, but how can you be absolutely sure the pattern continues in this way? What if this is like “Dots on a Circle,” and the obvious pattern breaks down after a few steps? You have to tie the pattern to the problem, so that it is clear why the pattern must continue in that way.
The first step in explaining a pattern is writing it down clearly. This brings us to another problem solving strategy.
Problem Solving Strategy 11 (Use a Variable!).
One of the most powerful tools we have is the use of a variable. If you find yourself doing calculations on things like “the number of squares,” or “the number of dots,” give those quantities a name! They become much easier to work with.
For now, just work on describing the pattern with variables.
- Stick with a 5 × 5 chess board for now, and consider a small square of size k × k . Describe the pattern: How many squares of size k × k fit on a chess board of size 5 × 5?
- What if the chess board is bigger? Based on the pattern above, how many squares of size k × k should fit on a chess board of size 10 × 10?
- What if you do not know how big the chess board is? Based on the pattern above, how many squares of size k × k should fit on a chess board of size n × n ?
Now comes the tough part: explaining the pattern. Let us focus on an 8 × 8 board. Since it measures 8 squares on each side, we can see that we get 8× 8 = 64 squares of size 1× 1. And since there is just a single board, we get just one square of size 8 × 8. But what about all the sizes in-between?
Using the Chess Board video in the previous chapter as a model, work with a partner to carefully explain why the number of 3 × 3 squares will be 6 · 6 = 36, and why the number of 4 × 4 squares will be 5 · 5 = 25.
There are many different explanations other than what is found in the video. Try to find your own explanation.
Here is what a final justification might look like ( watch the Chess Board video as a concrete example of this solution ):
- Solution (Chess Board Pattern).
-
Let n be the side of the chess board and let k be the side of the square. If the square is going to fit on the chess board at all, it must be true that k ≤ n . Otherwise, the square is too big.
If I put the k × k square in the upper left corner of the chess board, it takes up k spaces across and there are ( n – k ) spaces to the right of it. So I can slide the k × k square to the right ( n – k ) times, until it hits the top right corner of the chess board. The square is in ( n – k + 1) different positions, counting the starting position.
If I move the k × k square back to the upper left corner, I can shift it down one row and repeat the whole process again. Since there are ( n – k ) rows below the square, I can shift it down ( n – k ) times until it hits the bottom row. This makes ( n – k + 1) total rows that the square moves across, counting the top row.
So, there are ( n – k + 1) rows with ( n – k + 1) squares in each row. That makes ( n – k + 1) 2 total squares.
Thus, the solution is the sum of ( n – k + 1) 2 for all k ≤ n . In symbols:
\[\text{number of squares on an n} \times \text{n board} = \sum_{k = 1}^{n} (n - k + 1)^{2} \ldotp \nonumber \]
Once we are sure the pattern continues, we can use it to solve the problem. So go ahead!
- How many squares on a 10 × 10 chess board?
- What calculation would you do to solve that problem for a 100 × 100 chess board?
There is a number pattern that describes the number of pieces you get from the “Dots on a Circle” problem. If you want to solve the problem, go for it! Think about all of your problem solving strategies. But be sure that when you find a pattern, you can explain why it is the right pattern for this problem , and not just another pattern that seems to work but might not continue.


4.4: Functions of the Integumentary System
By the end of the section, you will be able to:
- Describe the different functions of the skin and the structures that enable them
- Explain how the skin helps maintain body temperature
The skin and accessory structures perform a variety of essential functions, such as protecting the body from invasion by microorganisms, chemicals, and other environmental factors; preventing dehydration; acting as a sensory organ; modulating body temperature and electrolyte balance; and synthesizing vitamin D. The underlying hypodermis has important roles in storing fats, forming a “cushion” over underlying structures, and providing insulation from cold temperatures.
Protection
The skin protects the rest of the body from the basic elements of nature such as wind, water, and UV sunlight. It acts as a protective barrier against water loss, due to the presence of layers of keratin and glycolipids in the stratum corneum. It also is the first line of defense against abrasive activity due to contact with grit, microbes, or harmful chemicals. Sweat excreted from sweat glands deters microbes from over-colonizing the skin surface by generating dermicidin, which has antibiotic properties.
Everyday Connection
Tattoos and Piercings
The word “armor” evokes several images. You might think of a Roman centurion or a medieval knight in a suit of armor. The skin, in its own way, functions as a form of armor—body armor. It provides a barrier between your vital, life-sustaining organs and the influence of outside elements that could potentially damage them.
For any form of armor, a breach in the protective barrier poses a danger. The skin can be breached when a child skins a knee or an adult has blood drawn—one is accidental and the other medically necessary. However, you also breach this barrier when you choose to “accessorize” your skin with a tattoo or body piercing. Because the needles involved in producing body art and piercings must penetrate the skin, there are dangers associated with the practice. These include allergic reactions; skin infections; blood-borne diseases, such as tetanus, hepatitis C, and hepatitis D; and the growth of scar tissue. Despite the risk, the practice of piercing the skin for decorative purposes has become increasingly popular. According to the American Academy of Dermatology, 24 percent of people from ages 18 to 50 have a tattoo.
Watch this video about tattooing . Tattooing has a long history, dating back thousands of years ago. The dyes used in tattooing typically derive from metals. A person with tattoos should be cautious when having a magnetic resonance imaging (MRI) scan because an MRI machine uses powerful magnets to create images of the soft tissues of the body, which could react with the metals contained in the tattoo dyes.
Sensory Function
You can feel an ant crawling on your skin, allowing you to flick it off before it bites, because the skin, and especially the hairs projecting from hair follicles in the skin, can sense changes in the environment. The hair root plexus surrounding the base of the hair follicle senses a disturbance, and then transmits the information to the central nervous system (brain and spinal cord), which can then respond by activating the skeletal muscles of your eyes to see the ant and the skeletal muscles of the body to act against the ant.
The skin acts as a sense organ because the epidermis, dermis, and the hypodermis contain specialized sensory nerve structures that detect touch, surface temperature, and pain. These receptors are more concentrated on the tips of the fingers, which are most sensitive to touch, especially the Meissner corpuscle (tactile corpuscle) (Figure \(\PageIndex{1}\)), which responds to light touch, and the Pacinian corpuscle (lamellated corpuscle), which responds to vibration. Merkel cells, seen scattered in the stratum basale, are also touch receptors. In addition to these specialized receptors, there are sensory nerves connected to each hair follicle, pain and temperature receptors scattered throughout the skin, and motor nerves innervating the arrector pili muscles and glands. This rich innervation helps you sense your environment and react accordingly.
Thermoregulation
The integumentary system helps regulate body temperature through its tight association with the sympathetic nervous system, the division of the nervous system involved in our fight-or-flight responses. The sympathetic nervous system is continuously monitoring body temperature and initiating appropriate motor responses. Recall that sweat glands, accessory structures to the skin, secrete water, salt, and other substances to cool the body when it becomes warm. Even when the body does not appear to be noticeably sweating, approximately 500 mL of sweat (insensible perspiration) are secreted a day. If the body becomes excessively warm due to high temperatures, vigorous activity (Figure \(\PageIndex{2.a-c}\)), or a combination of the two, sweat glands will be stimulated by the sympathetic nervous system to produce large amounts of sweat, as much as 0.7 to 1.5 L per hour for an active person. When the sweat evaporates from the skin surface, the body is cooled as body heat is dissipated.
In addition to sweating, arterioles (small blood vessels) in the dermis dilate so that excess heat carried by the blood can dissipate through the skin and into the surrounding environment (Figure \(\PageIndex{2.b}\)). This accounts for the skin redness that many people experience when exercising as the red pigment of the blood more visibly contributes to skin tone.
Conversely, when body temperatures drop, the arterioles constrict to minimize heat loss, particularly in the ends of the digits and tip of the nose. This reduced circulation can result in the skin taking on a paler hue. Although the temperature of the skin drops as a result, passive heat loss is prevented, and internal organs and structures remain warm. If the temperature of the skin drops too much (such as environmental temperatures below freezing), the conservation of body core heat can result in the skin actually freezing, a condition called frostbite.
Vitamin D Synthesis
The epidermal layer of human skin synthesizes the inactive form of vitamin D when exposed to UV radiation, which is eventually converted to calcitriol (the active chemical form of the vitamin) in the kidneys. Vitamin D is essential for normal absorption of calcium and phosphorus, which are required for healthy bones. The absence of sun exposure can lead to a lack of vitamin D in the body, leading to a condition called rickets , a painful condition in children where the bones are misshapen due to a lack of calcium, causing bowleggedness. Elderly individuals who suffer from vitamin D deficiency can develop a condition called osteomalacia, a softening of the bones. In present day society, vitamin D is added as a supplement to many foods, including milk and orange juice, compensating for the need for sun exposure.
In addition to its essential role in bone health, vitamin D is essential for general immunity against bacterial, viral, and fungal infections. Recent studies are also finding a link between insufficient vitamin D and cancer.
AGING AND THE.....
Integumentary System
All systems in the body accumulate subtle and some not-so-subtle changes as a person ages. Among these changes are reductions in cell division, metabolic activity, blood circulation, hormonal levels, and muscle strength (Figure \(\PageIndex{3}\)). In the skin, these changes are reflected in decreased mitosis in the stratum basale, leading to a thinner epidermis. The dermis, which is responsible for the elasticity and resilience of the skin, also exhibits a reduced ability to regenerate, which leads to slower wound healing. The hypodermis, with its fat stores, loses structure due to the reduction and redistribution of fat, which in turn contributes to the thinning and sagging of skin.
The accessory structures also have lowered activity, generating thinner hair and nails, and reduced amounts of sebum and sweat. A reduced sweating ability can cause some elderly to be intolerant to extreme heat. Other cells in the skin, such as melanocytes and dendritic cells, also become less active, leading to a paler skin tone and lowered immunity. Wrinkling of the skin occurs due to breakdown of its structure, which results from decreased collagen and elastin production in the dermis, weakening of muscles lying under the skin, and the inability of the skin to retain adequate moisture.
Many anti-aging products can be found in stores today. In general, these products try to rehydrate the skin and thereby fill out the wrinkles, and some stimulate skin growth using hormones and growth factors. Additionally, invasive techniques include collagen injections to plump the tissue and injections of BOTOX ® (the name brand of the botulinum neurotoxin) that paralyze the muscles that crease the skin and cause wrinkling.
Concept Review
The skin plays important roles in protection, sensing stimuli, thermoregulation, and vitamin D synthesis. It is the first layer of defense to prevent dehydration, infection, and injury to the rest of the body. Sweat glands in the skin allow the skin surface to cool when the body gets overheated. Thermoregulation is also accomplished by the dilation or constriction of heat-carrying blood vessels in the skin. Immune cells present among the skin layers patrol the areas to keep them free of foreign materials. Fat stores in the hypodermis aid in both thermoregulation and protection. Finally, the skin plays a role in the synthesis of vitamin D, which is necessary for our well-being but not easily available in natural foods.
Review Questions
Q. In humans, exposure of the skin to sunlight is required for ________.
A. vitamin D synthesis
B. arteriole constriction
C. folate production
D. thermoregulation
- Answer
-
Answer: A
Q. One of the functions of the integumentary system is protection. Which of the following does not directly contribute to that function?
A. stratum lucidum
B. desmosomes
C. folic acid synthesis
D. Merkel cells
- Answer
-
Answer: C
Q. An individual using a sharp knife notices a small amount of blood where he just cut himself. Which of the following layers of skin did he have to cut into in order to bleed?
A. stratum corneum
B. stratum basale
C. papillary dermis
D. stratum granulosum
- Answer
-
Answer: C
Q. As you are walking down the beach, you see a dead, dry, shriveled-up fish. Which layer of your epidermis keeps you from drying out?
A. stratum corneum
B. stratum basale
C. stratum spinosum
D. stratum granulosum
- Answer
-
Answer: A
Q. If you cut yourself and bacteria enter the wound, which of the following cells would help get rid of the bacteria?
A. Merkel cells
B. keratinocytes
C. Langerhans cells
D. melanocytes
- Answer
-
Answer: C
Critical Thinking Questions
Q. Why do people sweat excessively when exercising outside on a hot day?
- Answer
-
A. Sweating cools the body when it becomes warm. When the body temperature rises, such as when exercising on a hot day, the dermal blood vessels dilate, and the sweat glands begin to secrete more sweat. The evaporation of the sweat from the surface of the skin cools the body by dissipating heat.
Q. Explain your skin’s response to a drop in body core temperature.
- Answer
-
A. When the core body temperature drops, the body switches to heat-conservation mode. This can include an inhibition to excessive sweating and a decrease of blood flow to the papillary layers of the skin. This reduction of blood flow helps conserve body heat.
References
American Academy of Dermatology (US). Tattoos and body piercings [Internet]. Schaumburg, IL; c2013 [cited 2012 Nov 1]. Available from: http://www.aad.org/media-resources/s...ody-piercings/ .
Glossary
- Meissner corpuscle
- (also, tactile corpuscle) receptor in the skin that responds to light touch
- Pacinian corpuscle
- (also, lamellated corpuscle) receptor in the skin that responds to vibration
- rickets
- disease in children caused by vitamin D deficiency, which leads to the weakening of bones
- vitamin D
- compound that aids absorption of calcium and phosphates in the intestine to improve bone health
Contributors and Attributions
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OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology