12.4: Rotation of Axes
- Identify nondegenerate conic sections given their general form equations.
- Use rotation of axes formulas.
- Write equations of rotated conics in standard form.
- Identify conics without rotating axes.
As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone . The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone (Figure \(\PageIndex{1}\)).
Figure \(\PageIndex{1}\): The nondegenerate conic sections
Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in Figure \(\PageIndex{2}\). A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.
Figure \(\PageIndex{2}\): Degenerate conic sections
Identifying Nondegenerate Conics in General Form
In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.
\(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\)
where \(A\), \(B\), and \(C\) are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.
You may notice that the general form equation has an \(xy\) term that we have not seen in any of the standard form equations. As we will discuss later, the \(xy\) term rotates the conic whenever \(B\) is not equal to zero.
| Conic Sections | Example |
|---|---|
| ellipse | \(4x^2+9y^2=1\) |
| circle | \(4x^2+4y^2=1\) |
| hyperbola | \(4x^2−9y^2=1\) |
| parabola | \(4x^2=9y\) or \(4y^2=9x\) |
| one line | \(4x+9y=1\) |
| intersecting lines | \((x−4)(y+4)=0\) |
| parallel lines | \((x−4)(x−9)=0\) |
| a point | \(4x^2+4y^2=0\) |
| no graph | \(4x^2+4y^2=−1\) |
A conic section has the general form
\[Ax^2+Bxy+Cy^2+Dx+Ey+F=0 \label{gen}\]
where \(A\), \(B\), and \(C\) are not all zero. Table \(\PageIndex{2}\) summarizes the different conic sections where \(B=0\), and \(A\) and \(C\) are nonzero real numbers. This indicates that the conic has not been rotated.
| ellipse | \(Ax^2+Cy^2+Dx+Ey+F=0\), \(A≠C\) and \(AC>0\) |
|---|---|
| circle | \(Ax^2+Cy^2+Dx+Ey+F=0\), \(A=C\) |
| hyperbola | \(Ax^2−Cy^2+Dx+Ey+F=0\) or \(−Ax^2+Cy^2+Dx+Ey+F=0\), where \(A\) and \(C\) are positive |
| parabola | \(Ax^2+Dx+Ey+F=0\) or \(Cy^2+Dx+Ey+F=0\) |
- Rewrite the equation in the general form (Equation \ref{gen}), \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\)
-
Identify the values of \(A\) and \(C\) from the general form.
- If \(A\) and \(C\) are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.
- If \(A\) and \(C\) are equal and nonzero and have the same sign, then the graph may be a circle.
- If \(A\) and \(C\) are nonzero and have opposite signs, then the graph may be a hyperbola.
- If either \(A\) or \(C\) is zero, then the graph may be a parabola.
If \(B=0\), the conic section will have a vertical and/or horizontal axes. If \(B\) does not equal 0, as shown below, the conic section is rotated. Notice the phrase “may be” in the definitions. That is because the equation may not represent a conic section at all, depending on the values of \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\). For example, the degenerate case of a circle or an ellipse is a point:
\[Ax^2+By^2=0,\]
when \(A\) and \(B\) have the same sign.
The degenerate case of a hyperbola is two intersecting straight lines: \(Ax^2+By^2=0\), when \(A\) and \(B\) have opposite signs.
On the other hand, the equation, \(Ax^2+By^2+1=0\), when \(A\) and \(B\) are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.
Identify the graph of each of the following nondegenerate conic sections.
- \(4x^2−9y^2+36x+36y−125=0\)
- \(9y^2+16x+36y−10=0\)
- \(3x^2+3y^2−2x−6y−4=0\)
- \(−25x^2−4y^2+100x+16y+20=0\)
Solution
- Rewriting the general form (Equation \ref{gen}), we have \[\begin{align*} \color{red}{A} \color{black}x ^ { 2 } + \color{blue}{B} \color{black}x y + \color{red}{C} \color{black} y ^ { 2 } + \color{blue}{D} \color{black} x + \color{blue}{E} \color{black} y + \color{blue}{F} \color{black} &= 0 \\[4pt] 4 x ^ { 2 } + 0 x y + ( - 9 ) y ^ { 2 } + 36 x + 36 y + ( - 125 ) &= 0 \end{align*}\] with \(A=4\) and \(C=−9\), so we observe that \(A\) and \(C\) have opposite signs. The graph of this equation is a hyperbola.
- Rewriting the general form (Equation \ref{gen}), we have \[\begin{align*} \color{red}{A} \color{black}x ^ { 2 } + \color{blue}{B} \color{black}x y + \color{red}{C} \color{black} y ^ { 2 } + \color{blue}{D} \color{black} x + \color{blue}{E} \color{black} y + \color{blue}{F} \color{black} &= 0 \\[4pt] 0 x ^ { 2 } + 0 x y + 9 y ^ { 2 } + 16 x + 36 y + ( - 10 ) &= 0 \end{align*}\] with \(A=0\) and \(C=9\). We can determine that the equation is a parabola, since \(A\) is zero.
- Rewriting the general form (Equation \ref{gen}), we have \[\begin{align*} \color{red}{A} \color{black}x ^ { 2 } + \color{blue}{B} \color{black}x y + \color{red}{C} \color{black} y ^ { 2 } + \color{blue}{D} \color{black} x + \color{blue}{E} \color{black} y + \color{blue}{F} \color{black} &= 0 \\[4pt] 3 x ^ { 2 } + 0 x y + 3 y ^ { 2 } + ( - 2 ) x + ( - 6 ) y + ( - 4 ) &= 0 \end{align*}\] with \(A=3\) and \(C=3\). Because \(A=C\), the graph of this equation is a circle.
- Rewriting the general form (Equation \ref{gen}), we have \[\begin{align*} \color{red}{A} \color{black}x ^ { 2 } + \color{blue}{B} \color{black}x y + \color{red}{C} \color{black} y ^ { 2 } + \color{blue}{D} \color{black} x + \color{blue}{E} \color{black} y + \color{blue}{F} \color{black} &= 0 \\[4pt] ( - 25 ) x ^ { 2 } + 0 x y + ( - 4 ) y ^ { 2 } + 100 x + 16 y + 20 &= 0 \end{align*}\] with \(A=−25\) and \(C=−4\). Because \(AC>0\) and \(A≠C\), the graph of this equation is an ellipse.
Identify the graph of each of the following nondegenerate conic sections.
- \(16y^2−x^2+x−4y−9=0\)
- \(16x^2+4y^2+16x+49y−81=0\)
- Answer a
-
hyperbola
- Answer b
-
ellipse
Finding a New Representation of the Given Equation after Rotating through a Given Angle
Until now, we have looked at equations of conic sections without an \(xy\) term, which aligns the graphs with the x - and y -axes. When we add an \(xy\) term, we are rotating the conic about the origin. If the x - and y -axes are rotated through an angle, say \(\theta\),then every point on the plane may be thought of as having two representations: \((x,y)\) on the Cartesian plane with the original x -axis and y -axis, and \((x^\prime ,y^\prime )\) on the new plane defined by the new, rotated axes, called the x' -axis and y' -axis (Figure \(\PageIndex{3}\)).
Figure \(\PageIndex{3}\): The graph of the rotated ellipse \(x^2+y^2–xy–15=0\)
We will find the relationships between \(x\) and \(y\) on the Cartesian plane with \(x^\prime \) and \(y^\prime \) on the new rotated plane (Figure \(\PageIndex{4}\)).
Figure \(\PageIndex{4}\): The Cartesian plane with \(x\)- and \(y\)-axes and the resulting \(x^\prime\)− and \(y^\prime\)−axes formed by a rotation by an angle \(\theta\).
The original coordinate x - and y -axes have unit vectors \(\hat{i}\) and \(\hat{j}\). The rotated coordinate axes have unit vectors \(\hat{i}^\prime\) and \(\hat{j}^\prime\).The angle \(\theta\) is known as the angle of rotation (Figure \(\PageIndex{5}\)). We may write the new unit vectors in terms of the original ones.
\[\hat{i}′=\cos \theta \hat{i}+\sin \theta \hat{j}\]
\[\hat{j}′=−\sin \theta \hat{i}+\cos \theta \hat{j}\]
Figure \(\PageIndex{5}\): Relationship between the old and new coordinate planes.
Consider a vector \(\vec{u}\) in the new coordinate plane. It may be represented in terms of its coordinate axes.
\[\begin{align*} \vec{u}&=x^\prime i′+y^\prime j′ \\[4pt] &=x^\prime (i \cos \theta+j \sin \theta)+y^\prime (−i \sin \theta+j \cos \theta) & \text{Substitute.} \\[4pt] &=ix' \cos \theta+jx' \sin \theta−iy' \sin \theta+jy' \cos \theta & \text{Distribute.} \\[4pt] &=ix' \cos \theta−iy' \sin \theta+jx' \sin \theta+jy' \cos \theta & \text{Apply commutative property.} \\[4pt] &=(x' \cos \theta−y' \sin \theta)i+(x' \sin \theta+y' \cos \theta)j & \text{Factor by grouping.} \end{align*}\]
Because \(\vec{u}=x^\prime i′+y^\prime j′\), we have representations of \(x\) and \(y\) in terms of the new coordinate system.
\(x=x^\prime \cos \theta−y^\prime \sin \theta\)
and
\(y=x^\prime \sin \theta+y^\prime \cos \theta\)
If a point \((x,y)\) on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle \(\theta\) from the positive x -axis, then the coordinates of the point with respect to the new axes are \((x^\prime ,y^\prime )\). We can use the following equations of rotation to define the relationship between \((x,y)\) and \((x^\prime , y^\prime )\):
\[x=x^\prime \cos \theta−y^\prime \sin \theta\]
and
\[y=x^\prime \sin \theta+y^\prime \cos \theta\]
- Find \(x\) and \(y\) where \(x=x^\prime \cos \theta−y^\prime \sin \theta\) and \(y=x^\prime \sin \theta+y^\prime \cos \theta\).
- Substitute the expression for \(x\) and \(y\) into in the given equation, then simplify.
- Write the equations with \(x^\prime \) and \(y^\prime \) in standard form.
Find a new representation of the equation \(2x^2−xy+2y^2−30=0\) after rotating through an angle of \(\theta=45°\).
Solution
Find \(x\) and \(y\), where \(x=x^\prime \cos \theta−y^\prime \sin \theta\) and \(y=x^\prime \sin \theta+y^\prime \cos \theta\).
Because \(\theta=45°\),
\[\begin{align*} x &= x^\prime \cos(45°)−y^\prime \sin(45°) \\[4pt] x &= x^\prime \left(\dfrac{1}{\sqrt{2}}\right)−y^\prime \left(\dfrac{1}{\sqrt{2}}\right) \\[4pt] x &=\dfrac{x^\prime −y^\prime }{\sqrt{2}} \end{align*}\]
and
\[\begin{align*} y &= x^\prime \sin(45°)+y^\prime \cos(45°) \\[4pt] y &= x^\prime \left(\dfrac{1}{\sqrt{2}}\right) + y^\prime \left(\dfrac{1}{\sqrt{2}}\right) \\[4pt] y &= \dfrac{x^\prime +y^\prime }{\sqrt{2}} \end{align*}\]
Substitute \(x=x^\prime \cos\theta−y^\prime \sin\theta\) and \(y=x^\prime \sin \theta+y^\prime \cos \theta\) into \(2x^2−xy+2y^2−30=0\).
\(2{\left(\dfrac{x^\prime −y^\prime }{\sqrt{2}}\right)}^2−\left(\dfrac{x^\prime −y^\prime }{\sqrt{2}}\right)\left(\dfrac{x^\prime +y^\prime }{\sqrt{2}}\right)+2{\left(\dfrac{x^\prime +y^\prime }{\sqrt{2}}\right)}^2−30=0\)
Simplify .
\(\begin{array}{rl} 2\dfrac{(x^\prime−y^\prime )(x^\prime −y^\prime )}{2}−\dfrac{(x^\prime −y^\prime )(x^\prime +y^\prime )}{2}+2\dfrac{(x^\prime +y^\prime )(x^\prime +y^\prime )}{2}−30=0 & \text{FOIL method} \\[4pt] {x^\prime }^2−2x^\prime y^\prime +{y^\prime }^2−\dfrac{({x^\prime }^2−{y^\prime }^2)}{2}+{x^\prime }^2+2x^\prime y^\prime +{y^\prime }^2−30=0 & \text{Combine like terms.} \\[4pt] 2{x^\prime }^2+2{y^\prime }^2−\dfrac{({x^\prime }^2−{y^\prime }^2)}{2}=30 & \text{Combine like terms.} \\[4pt] 2(2{x^\prime }^2+2{y^\prime }^2−\dfrac{({x^\prime }^2−{y^\prime }^2)}{2})=2(30) & \text{Multiply both sides by 2.} \\[4pt] 4{x^\prime }^2+4{y^\prime }^2−({x^\prime }^2−{y^\prime }^2)=60 & \text{Simplify. } \\[4pt] 4{x^\prime }^2+4{y^\prime }^2−{x^\prime }^2+{y^\prime }2=60 & \text{Distribute.} \\[4pt] \dfrac{3{x^\prime }^2}{60}+\dfrac{5{y^\prime }^2}{60}=\dfrac{60}{60} & \text{Set equal to 1.} \end{array} \)
Write the equations with \(x^\prime \) and \(y^\prime \) in the standard form.
\[\dfrac{{x^\prime }^2}{20}+\dfrac{{y^\prime}^2}{12}=1 \nonumber\]
This equation is an ellipse. Figure \(\PageIndex{6}\) shows the graph.
Figure \(\PageIndex{6}\)
Writing Equations of Rotated Conics in Standard Form
Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\) into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the \(x^\prime \) and \(y^\prime \) coordinate system without the \(x^\prime y^\prime \) term, by rotating the axes by a measure of \(\theta\) that satisfies
\[\cot(2\theta)=\dfrac{A−C}{B}\]
We have learned already that any conic may be represented by the second degree equation
\(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\)
where \(A\), \(B\),and \(C\) are not all zero. However, if \(B≠0\), then we have an \(xy\) term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle \(\theta\) where \(\cot(2\theta)=\dfrac{A−C}{B}\).
- If \(\cot(2\theta)>0\), then \(2\theta\) is in the first quadrant, and \(\theta\) is between \((0°,45°)\).
- If \(\cot(2\theta)<0\), then \(2\theta\) is in the second quadrant, and \(\theta\) is between \((45°,90°)\).
- If \(A=C\), then \(\theta=45°\).
- Find \(\cot(2\theta)\).
- Find \(\sin \theta\) and \(\cos \theta\).
- Substitute \(\sin \theta\) and \(\cos \theta\) into \(x=x^\prime \cos \theta−y^\prime \sin \theta\) and \(y=x^\prime \sin \theta+y^\prime \cos \theta\).
- Substitute the expression for \(x\) and \(y\) into in the given equation, and then simplify.
- Write the equations with \(x^\prime \) and \(y^\prime \) in the standard form with respect to the rotated axes.
Rewrite the equation \(8x^2−12xy+17y^2=20\) in the \(x^\prime y^\prime \) system without an \(x^\prime y^\prime \) term.
Solution
First, we find \(\cot(2\theta)\).
\(8x^2−12xy+17y^2=20\rightarrow A=8\), \(B=−12\) and \(C=17\)
Figure \(\PageIndex{7}\)
From Figure \(\PageIndex{7}\):
\[ \begin{align*} \cot(2\theta) &=\dfrac{A−C}{B}=\dfrac{8−17}{−12} \\[4pt] & =\dfrac{−9}{−12}=\dfrac{3}{4} \end{align*}\]
\(\cot(2\theta)=\dfrac{3}{4}=\dfrac{\text{adjacent}}{\text{opposite}}\)
So the hypotenuse is
\[ \begin{align*} 3^2+4^2 &=h^2 \\[4pt] 9+16 &=h^2 \\[4pt] 25&=h^2 \\[4pt] h&=5 \end{align*}\]
Next, we find \(\sin \theta\) and \(\cos \theta\).
\[ \begin{align*} \sin \theta &=\sqrt{\dfrac{1−\cos(2\theta)}{2}}=\sqrt{\dfrac{1−\dfrac{3}{5}}{2}}=\sqrt{\dfrac{\dfrac{5}{5}−\dfrac{3}{5}}{2}}=\sqrt{\dfrac{5−3}{5}⋅\dfrac{1}{2}}=\sqrt{\dfrac{2}{10}}=\sqrt{\dfrac{1}{5}} \\ \sin \theta &= \dfrac{1}{\sqrt{5}} \\ \cos \theta &= \sqrt{\dfrac{1+\cos(2\theta)}{2}}=\sqrt{\dfrac{1+\dfrac{3}{5}}{2}}=\sqrt{\dfrac{\dfrac{5}{5}+\dfrac{3}{5}}{2}}=\sqrt{\dfrac{5+3}{5}⋅\dfrac{1}{2}}=\sqrt{\dfrac{8}{10}}=\sqrt{\dfrac{4}{5}} \\ \cos \theta &= \dfrac{2}{\sqrt{5}} \end{align*}\]
Substitute the values of \(\sin \theta\) and \(\cos \theta\) into \(x=x^\prime \cos \theta−y^\prime \sin \theta\) and \(y=x^\prime \sin \theta+y^\prime \cos \theta\).
\[ \begin{align*} x &=x'\cos \theta−y^\prime \sin \theta \\[4pt] &=x^\prime \left(\dfrac{2}{\sqrt{5}}\right)−y^\prime \left(\dfrac{1}{\sqrt{5}}\right) \\[4pt] &=\dfrac{2x^\prime −y^\prime }{\sqrt{5}} \end{align*}\]
and
\[ \begin{align*} y&=x^\prime \sin \theta+y^\prime \cos \theta \\[4pt] &=x^\prime \left(\dfrac{1}{\sqrt{5}}\right)+y^\prime \left(\dfrac{2}{\sqrt{5}}\right) \\[4pt] &=\dfrac{x^\prime +2y^\prime }{\sqrt{5}} \end{align*}\]
Substitute the expressions for \(x\) and \(y\) into in the given equation, and then simplify.
\[ \begin{align*} 8{\left(\dfrac{2x^\prime −y^\prime }{\sqrt{5}}\right)}^2−12\left(\dfrac{2x^\prime −y^\prime }{\sqrt{5}}\right)\left(\dfrac{x^\prime +2y^\prime }{\sqrt{5}}\right)+17{\left(\dfrac{x^\prime +2y^\prime }{\sqrt{5}}\right)}^2&=20 \\[4pt] 8\left(\dfrac{(2x^\prime −y^\prime )(2x^\prime −y^\prime )}{5}\right)−12\left(\dfrac{(2x^\prime −y^\prime )(x^\prime +2y^\prime )}{5}\right)+17\left(\dfrac{(x^\prime +2y^\prime )(x^\prime +2y^\prime )}{5}\right)&=20 \\[4pt] 8(4{x^\prime }^2−4x^\prime y^\prime +{y^\prime }^2)−12(2{x^\prime }^2+3x^\prime y^\prime −2{y^\prime }^2)+17({x^\prime }^2+4x^\prime y^\prime +4{y^\prime }^2)&=100 \\[4pt] 32{x^\prime }^2−32x^\prime y^\prime +8{y^\prime }^2−24{x^\prime }^2−36x^\prime y^\prime +24{y^\prime }^2+17{x^\prime }^2+68x^\prime y^\prime +68{y^\prime }^2&=100 \\[4pt] 25{x^\prime }^2+100{y^\prime }^2&=100 \\[4pt] \dfrac{25}{100}{x^\prime }^2+\dfrac{100}{100}{y^\prime }^2&=\dfrac{100}{100} \end{align*}\]
Write the equations with \(x^\prime \) and \(y^\prime \) in the standard form with respect to the new coordinate system.
\[\dfrac{{x^\prime }^2}{4}+\dfrac{{y^\prime }^2}{1}=1 \nonumber\]
Figure \(\PageIndex{8}\) shows the graph of the ellipse.
Figure \(\PageIndex{8}\)
Rewrite the \(13x^2−6\sqrt{3}xy+7y^2=16\) in the \(x^\prime y^\prime \) system without the \(x^\prime y^\prime \) term.
- Answer
-
\(\dfrac{{x^\prime }^2}{4}+\dfrac{{y^\prime }^2}{1}=1\)
Graph the following equation relative to the \(x^\prime y^\prime \) system:
\(x^2+12xy−4y^2=30\)
Solution
First, we find \(\cot(2\theta)\).
\(x^2+12xy−4y^2=20\rightarrow A=1\), \(B=12\),and \(C=−4\)
\[\begin{align*} \cot(2\theta) &= \dfrac{A−C}{B} \\ \cot(2\theta) &= \dfrac{1−(−4)}{12} \\ \cot(2\theta) &= \dfrac{5}{12} \end{align*}\]
Because \(\cot(2\theta)=\dfrac{5}{12}\), we can draw a reference triangle as in Figure \(\PageIndex{9}\).
Figure \(\PageIndex{9}\)
\(\cot(2\theta)=\dfrac{5}{12}=\dfrac{adjacent}{opposite}\)
Thus, the hypotenuse is
\[ \begin{align*} 5^2+{12}^2&=h^2 \\[4pt] 25+144 &=h^2 \\[4pt] 169 &=h^2 \\[4pt] h&=13 \end{align*}\]
Next, we find \(\sin \theta\) and \(\cos \theta\). We will use half-angle identities.
\(\sin \theta=\sqrt{\dfrac{1−\cos(2\theta)}{2}}=\sqrt{\dfrac{1−\dfrac{5}{13}}{2}}=\sqrt{\dfrac{\dfrac{13}{13}−\dfrac{5}{13}}{2}}=\sqrt{\dfrac{8}{13}⋅\dfrac{1}{2}}=\dfrac{2}{\sqrt{13}}\)
\(\cos \theta=\sqrt{\dfrac{1+\cos(2\theta)}{2}}=\sqrt{\dfrac{1+\dfrac{5}{13}}{2}}=\sqrt{\dfrac{\dfrac{13}{13}+\dfrac{5}{13}}{2}}=\sqrt{\dfrac{18}{13}⋅\dfrac{1}{2}}=\dfrac{3}{\sqrt{13}}\)
Now we find \(x\) and \(y\).
\(x=x^\prime \cos \theta−y^\prime \sin \theta\)
\(x=x^\prime \left(\dfrac{3}{\sqrt{13}}\right)−y^\prime \left(\dfrac{2}{\sqrt{13}}\right)\)
\(x=\dfrac{3x^\prime −2y^\prime }{\sqrt{13}}\)
and
\(y=x^\prime \sin \theta+y^\prime \cos \theta\)
\(y=x^\prime \left(\dfrac{2}{\sqrt{13}}\right)+y^\prime \left(\dfrac{3}{\sqrt{13}}\right)\)
\(y=\dfrac{2x^\prime +3y^\prime }{\sqrt{13}}\)
Now we substitute \(x=\dfrac{3x^\prime −2y^\prime }{\sqrt{13}}\) and \(y=\dfrac{2x^\prime +3y^\prime }{\sqrt{13}}\) into \(x^2+12xy−4y^2=30\).
\(\begin{array}{rl} {\left(\dfrac{3x^\prime −2y^\prime }{\sqrt{13}}\right)}^2+12\left(\dfrac{3x^\prime −2y^\prime }{\sqrt{13}}\right)\left(\dfrac{2x^\prime +3y^\prime }{\sqrt{13}}\right)−4{\left(\dfrac{2x^\prime +3y^\prime }{\sqrt{13}}\right)}^2=30 \\ \left(\dfrac{1}{13}\right)[ {(3x^\prime −2y^\prime )}^2+12(3x^\prime −2y^\prime )(2x^\prime +3y^\prime )−4{(2x^\prime +3y^\prime )}^2 ]=30 & \text{Factor.} \\ \left(\dfrac{1}{13}\right)[ 9{x^\prime }^2−12x^\prime y^\prime +4{y^\prime }^2+12(6{x^\prime }^2+5x^\prime y^\prime −6{y^\prime }^2)−4(4{x^\prime }^2+12x^\prime y^\prime +9{y^\prime }^2) ]=30 & \text{Multiply.} \\ \left(\dfrac{1}{13}\right)[ 9{x^\prime }^2−12x^\prime y^\prime +4{y^\prime }^2+72{x^\prime }^2+60x^\prime y^\prime −72{y^\prime }^2−16{x^\prime }^2−48x^\prime y^\prime −36{y^\prime }^2 ]=30 & \text{Distribute.} \\ \left(\dfrac{1}{13}\right)[ 65{x^\prime }^2−104{y^\prime }^2 ]=30 & \text{Combine like terms.} \\ 65{x^\prime }^2−104{y^\prime }^2=390 & \text{Multiply.} \\ \dfrac{{x^\prime }^2}{6}−\dfrac{4{y^\prime }^2}{15}=1 & \text{Divide by 390.} \end{array}\)
Figure \(\PageIndex{10}\) shows the graph of the hyperbola \(\dfrac{{x^\prime }^2}{6}−\dfrac{4{y^\prime }^2}{15}=1\)
Figure \(\PageIndex{10}\)
Identifying Conics without Rotating Axes
Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is
\(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\)
If we apply the rotation formulas to this equation we get the form
\(A′{x^\prime }^2+B′x^\prime y^\prime +C′{y^\prime }^2+D′x^\prime +E′y^\prime +F′=0\)
It may be shown that
\(B^2−4AC={B′}^2−4A′C′\)
The expression does not vary after rotation, so we call the expression invariant . The discriminant, \(B^2−4AC\), is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.
If the equation
\[Ax^2+Bxy+Cy^2+Dx+Ey+F=0\]
is transformed by rotating axes into the equation
\[A′{x^\prime }^2+B′x^\prime y^\prime +C′{y^\prime }^2+D′x^\prime +E′y^\prime +F′=0\]
then \[B^2−4AC={B′}^2−4A′C′\]
The equation \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\) is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these. If the discriminant, \(B^2−4AC\), is
- \(<0\), the conic section is an ellipse
- \(=0\), the conic section is a parabola
- \(>0\), the conic section is a hyperbola
Identify the conic for each of the following without rotating axes.
- \(5x^2+2\sqrt{3}xy+2y^2−5=0\)
- \(5x^2+2\sqrt{3}xy+12y^2−5=0\)
Solution
a. Let’s begin by determining \(A\), \(B\), and \(C\).
\(\underbrace{5}_{A}x^2+\underbrace{2\sqrt{3}}_{B}xy+\underbrace{2}_{C}y^2−5=0\)
Now, we find the discriminant.
\[\begin{align*} B^2−4AC &= {(2\sqrt{3})}^2−4(5)(2) \\ &=4(3)−40 \\ &=12−40 \\ &=−28<0 \end{align*}\]
Therefore, \(5x^2+2\sqrt{3}xy+2y^2−5=0\) represents an ellipse.
b. Again, let’s begin by determining \(A\),\(B\), and \(C\).
\(\underbrace{5}_{A}x^2+\underbrace{2\sqrt{3}}_{B}xy+\underbrace{12}_{C}y^2−5=0 \nonumber\)
Now, we find the discriminant.
\[\begin{align*} B^2−4AC &= {(2\sqrt{3})}^2−4(5)(12) \\ &= 4(3)−240 \\ &= 12−240 \\ &=−228<0 \end{align*}\]
Therefore, \(5x^2+2\sqrt{3}xy+12y^2−5=0\) represents an ellipse.
Identify the conic for each of the following without rotating axes.
- \(x^2−9xy+3y^2−12=0\)
- \(10x^2−9xy+4y^2−4=0\)
- Answer a
-
hyperbola
- Answer b
-
ellipse
Key Equations
| General Form equation of a conic section | \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\) |
| Rotation of a conic section |
\(x=x^\prime \cos \theta−y^\prime \sin \theta\) \(y=x^\prime \sin \theta+y^\prime \cos \theta\) |
| Angle of rotation | \(\theta\), where \(\cot(2\theta)=\dfrac{A−C}{B}\) |
Key Concepts
- Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
- A nondegenerate conic section has the general form \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\) where \(A\), \(B\) and \(C\) are not all zero. The values of \(A\), \(B\), and \(C\) determine the type of conic. See Example \(\PageIndex{1}\).
- Equations of conic sections with an \(xy\) term have been rotated about the origin. See Example \(\PageIndex{2}\).
- The general form can be transformed into an equation in the \(x^\prime \) and \(y^\prime \) coordinate system without the \(x^\prime y^\prime \) term. See Example \(\PageIndex{3}\) and Example \(\PageIndex{4}\).
- An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See Example \(\PageIndex{5}\).


4.6: Editing Focus- More on Characterization and Point of View
By the end of this section, you will be able to:
- Correctly identify and use conventions of the personal narrative genre, including structure, paragraphing, tone, and mechanics.
- Gain experience negotiating variations in genre conventions.
When you began writing your story, you likely concentrated on developing the plot and placing events and ideas in an order that made sense. Now, as you edit, work toward refining characterization and point of view in your narrative.
The Storyteller's Personas: Characterization
After you have developed your narrative, consider the characterization you used to develop the narrator’s voice. Characterization helps create a picture of the narrator as a character, drawing the reader in and giving life and interest to the story. Persona , from the Latin word for a type of mask worn by stage actors, develops the impression the narrator leaves on the reader.
A persona is actually a construct. Although the story is about you, the actual character on the page is not you. Like the Latin word, it is a mask, or a stand-in for you. You can carefully choose a persona for your character to communicate more authentically and directly with your audience. In real life, your persona refers to your appearance, mannerisms, voice, and body language. In a personal narrative, it is similar—the voice of the narrator as developed through thoughts, actions, and dialogue. Just as you might have a different persona when interacting with a teacher than you would with a friend, your persona as narrator should be carefully crafted to create the most meaning.  You can be who you want (or do not want) to be.
Consider how Twain creates a persona at the beginning of the excerpt from Life on the Mississippi . The opening of the excerpt provides information about both the narrator and Mr. Bixby, and the characterization of each supports the other. The narrator is portrayed as the junior steersman. He is confident but perhaps slightly cross at doing the majority of the work while the more experienced Mr. Bixby plays “gentleman of leisure.
I had become a good steersman; so good, indeed, that I had all the work to do on our watch, night and day; Mr. Bixby seldom made a suggestion to me; all he ever did was to take the wheel on particularly bad nights or in particularly bad crossings, land the boat when she needed to be landed, play gentleman of leisure nine-tenths of the watch, and collect the wages.
Like Twain, you can develop persona through characterization in your story. Keep in mind that characters are viewed through your eyes. Other people or characters may view them differently, but in your personal narrative, it is your opinion that counts most. Use the following tips to strengthen characterization:
- Observe and report on a character’s surroundings.
- Relay a character’s personality through mannerisms, style, physical appearance, and dialogue.
- Build characters on the basis of relationships and roles.
- Report on how characters interact with one another and how they confront events.
- Relate characters to time.
- Express a character’s role in plot events.
Choosing the First-Person Point of View
Remember that point of view has to do with the narrator of a story and how that narrator reports and understands events. A narrator may be biased or only have access to certain information and will relate and interpret events accordingly. You most likely have chosen or been assigned to write from the first-person point of view.
First-Person Point of View
A first-person narrator is a character in the story, whether fiction or nonfiction. Memoirs and autobiographies, as well as personal narratives and many works of fiction, are narrated from the first-person point of view. In telling their stories, first-person narrators, whether real people or fictional characters, are part of the story. They participate in the action, share opinions, and provide descriptions and interpretations. It is important in writing to remember that these narrators know only what they observe around them—what they learn from dialogue, what they are told, and what actions or events occur. They do not know what other characters are thinking and cannot go beyond what they imagine regarding other characters. For example, as Mark Twain begins to steer the boat, he observes Mr. Bixby leave and the others arrive. He cannot know what they are thinking or what motivates them. First-person narrators use I and me to indicate that they are the ones speaking and observing.
First-person point of view is the most frequently used in memoirs and personal narratives and serves as an authentic and credible point of view.
Third-Person Point of View
Third-person narration is also used frequently in narrative writing, but usually in fiction and nonpersonal narratives. In this point of view, a narrator who is not a character tells the story. In other words, the narrator is outside the story and sees it from a broader angle. The narrator’s point of view may be limited point of view , in which case the narrator aligns with one or several characters and knows only what they know—that is, the narrator reveals only the thoughts of that one or those several characters. Alternatively, the point of view may be omniscient , or all-knowing. Omniscient narrators know all characters’ thoughts and actions regardless of whether the characters are present. Third-person narrators do not put themselves in the story, and they narrate with third-person pronouns such as he , she , and they . This point of view may provide more reliable and objective narration—but not always.
Second-Person Point of View
A final type of narration is the second-person point of view , in which the narrator uses the pronoun you to address readers directly. As in first-person narration, the narrator in this case is usually a character in the story; however, second-person narration folds the reader into the story as a character, a technique that draws them closer to the plot. Consider how American author Nathaniel Hawthorne (1804–1864) draws in the reader in the opening of his short story “The Haunted Mind” (1835).
What a singular moment is the first one when you have hardly begun to recollect yourself after starting from midnight slumber! By unclosing your eyes so suddenly, you seem to have surprised the personages of your dream in full convocation round your bed, and catch one broad glance at them before they can flit into obscurity.
The second-person point of view is used much less in literary writing than either first or third person. Although it sometimes can be an effective perspective for experienced authors, second person presents problems for both readers and writers. It is difficult to develop successfully, and it’s easy to lose track of the narration, thus confusing the reader. You can be sure that if you as the author have difficulty following the narration as you work through the revision process, your readers have little chance. Also, use of the second-person point of view may not be clear to readers who might not notice a difference between addressing readers as you simply to provide some information and actually drawing readers into the story as characters. Finally, second person can be difficult for readers to trust, for you are essentially asking them to suspend disbelief and take on all the qualities and experiences that you, as the author, assign them.
You will mostly likely use first-person narration as you retell a turning point in your life, but whichever narration you choose, be consistent.
Practice with Point of View
To become more familiar with first- and third-person points of view, rewrite a paragraph of your personal narrative using both types of third-person point of view, limited and omniscient. For the limited point of view, choose one of the characters for the narrator to focus on. For the omniscient point of view, focus on all of the characters. Then reflect on how the point of view changes the story. Which point of view do you prefer—first, limited third, or omniscient third? Why?


4.7: Topics for Discussion
- Think of a recent infrastructure development near where you live, such as a recently-built road or dam. Try to come up with a list of ecosystem services that were damaged by this development. Who carries the costs of these lost services? Do you think the benefits from the development were worth the costs? Explain your answer.
- Do individual organisms, populations, species, and biological communities have rights? What about physical features such as lakes, rivers, and mountains? While explaining your answer, also think about where we should draw the line of moral responsibility in how we care for nature.
- A European botanist on holiday visits your area. During a short hike, you show this botanist a plant used as a traditional treatment for malaria. The botanist takes samples of this plant back to Europe, where subsequent testing shows that it can be used to develop an effective anti-malarial drug. Who do you think should receive the profits from this new drug? The botanist who undertook the trip, and you because you showed the botanist the plant? What about the organization that funded the drug’s development, and the scientists who synthesised the new drug? What about all the people who educated you and your family in the plant’s value? If the profits belong to multiple entities, how should it be divided?
- More than a decade ago, the shark ecotourism industry at Gansbaai, South Africa, was estimated at US $4.4 million annually (Hara et al., 2003)—it has been increasing ever since. There are an estimated 900 great white sharks (Carcharodon carcharias, VU) living in Gansbaai (Towner et al., 2013). Assuming the average white shark lives for 70 years (Hamady et al., 2014), what is the value of each shark at Gansbaai? Can you find (or estimate) the price that a single shark sold for food would obtain on the world market? How do these values compare? What do you think is the best use of the sharks?
Allsopp, M.H., W.J. de Lange, and R. Veldtman. 2008. Valuing insect pollination services with cost of replacement. PLoS ONE 3: e3128. https://doi.org/10.1371/journal.pone.0003128 A study estimating the economic value of local pollination services.
Costanza, R., R. de Groot, P. Sutton, et al. 2014. Changes in the global value of ecosystem services. Global Environmental Change 26: 152–58. https://doi.org/10.1016/j.gloenvcha.2014.04.002 An attempt to value all ecosystem services.
Farber, S.C., R. Costanza, and A.M. Wilson. 2002. Economic and ecological concepts for valuing ecosystem services. Ecological Economics 41: 375–92. https://doi.org/10.1016/S0921-8009(02)00088-5 Methods for estimating the value of ecosystem services.
Isbell, F., V. Calcagno, A. Hector, et al. 2011. High plant diversity is needed to maintain ecosystem services. Nature 477: 199–202. https://doi.org/10.1038/nature10282 Maintaining ecosystem services requires protecting a diversity of species.
Koné, I., J.E. Lambert, J. Refisch, et al. 2008. Primate seed dispersal and its potential role in maintaining useful tree species in the Taï region, Côte d’Ivoire: Implications for the conservation of forest fragments. Tropical Conservation Science 1: 293–306. https://doi.org/10.1177%2F194008290800100309 Maintaining primate populations is important also for humans who rely on forest resources.
Markandya, A., T. Taylor, A. Longo, et al. 2008. Counting the cost of vulture decline—an appraisal of the human health and other benefits of vultures in India. Ecological Economics 67: 194-204. https://doi.org/10.1016/j.ecolecon.2008.04.020 A study illustrating the value of vultures.
Naidoo, R., B. Fisher, A. Manica, et al. 2016. Estimating economic losses to tourism in Africa from the illegal killing of elephants. Nature Communications 7: 13379. https://doi.org/10.1038/ncomms13379 Africa loses US $25 million annually from elephant poaching.
Peterson, G.D., Z.V. Harmackova, M. Meacham, et al. 2018. Welcoming different perspectives in IPBES: “Nature’s contributions to people” and “Ecosystem services”. Ecology and Society 23: 39. https://doi.org/10.5751/ES-10134-230139 Addressing shortcomings of the ecosystem services concept
Schleicher, J., M. Schaafsma, N.D. Burgess, et al. 2018. Poorer without it? The neglected role of the natural environment in poverty and wellbeing. Sustainable Development 25: 83–98. https://doi.org/10.1002/sd.1692 The environment and human well-being are intricately linked.


3.12: Describing Retailers and Channels
- Define single channel, multi-channel, and omnichannel retailing
Marketing channels, also known as distribution channels, are the ways that products or services get from the producer to the consumer. All of the activities in between are part of a product’s distribution channel. This includes:
- Logistics: assembly, warehousing, and transportation
- Facilitation: channel coordination, marketing, promotion, financing, and post-purchase service and maintenance
- Transaction buying and selling
These activities support the exchange of goods, transferring products and services to the consumer, and transferring payments back to the producer. Over time, only value-added activities remain part of a given channel to reduce costs and ensure efficiency.
Earlier, we described the distribution channel for a can of soup. After being produced, it will be warehoused either on-site at the manufacturing facility or at a central distribution center until it is ordered by a retailer. Once ordered, it will be shipped to their warehouse and bundled with other items to be trucked to an individual store. At the store, it will be unboxed and put on the shelf to be available for shoppers.
This example reflects not only the functional activities (logistics, facilitation and transaction), but also identifies groups involved:
- Producers (manufacturers, farmers or craftsmen)
- Intermediaries (warehousers, logistics providers, wholesalers, financiers or credit providers, retailers, etc.)
- Consumers (the individual or institution making the buying decision. Note that the focus is on the individual making the buying decision because the end users of some items, such as baby or pet care products, are not engaged in the transaction.)
It is worth nothing that not all groups are involved in each marketing channel. In fact, marketing channels are distinguished by how the individual groups manage logistics, facilitation and transaction. The four marketing channels are as follows:
- Direct: Producer to consumer
- Indirect: Producer to intermediary to consumer
- Dual: Distribution to consumers through multiple channels, often determined by the producer and their development of intermediaries as channel partners
- Reverse: User to beneficiary (Note: this is an uncommon channel and one that most applies when existing goods are recycled or repurposed to re-enter the market.)
In direct distribution channels, producers market and sell their products directly to the consumer. In doing so, the producers absorb the activities and costs associated with logistics, such as warehousing and distributing their goods. However, as a trade-off, the producer has greater control over the marketing, pricing and promotion of their items. Further, because they interact directly with the consumer, they can benefit from unfiltered feedback, allowing them to optimize products, services, promotional activity or other elements of the customer experience.
Direct channels are common in commercial settings, especially where products and services are customized for individual customers, like enterprise software or industrial components. Tesla is an example of a manufacturer using a direct channel in the consumer space, as they market their cars directly to consumers without going through a dealer network. Catalogue marketing is another good example of direct channels in consumer goods. Similarly, the internet serves as a direct channel that many brands have used to broaden their reach.
Indirect channels are very familiar to us because it’s what we most commonly associate with retail, especially in the grocery industry. In indirect channels, producers sell part of their inventory to a collection of wholesalers and retailers who market the product to consumers. While the inclusion of wholesalers and retailers as intermediaries does add some complexity, it also has critical benefits. It allows the individual producers and intermediaries to specialize, increasing efficiency and reducing cost. It also mitigates risk for producers because they don’t own the entirety of their inventory. For the intermediaries, they will realize greater revenue for the product than the producer would have, but they’ve also taken on some risk by buying the product.
Dual distribution describes any marketing arrangement where the manufacturer uses multiple channels to distribute a product. For instance, they may sell directly to end users as well as through other companies for resale. Think about Apple products, which can be purchased through a direct channel either on Apple’s website or at the Apple Store or through an indirect channel at a mass retailer like Target or specialty retailers like Verizon or AT&T stores.
Understanding channels, their implicit activities and their actors, is important because it helps us understand:
- How products or services get to the end user and payments are transferred back to the producer
- How channel partners mitigate risk and add value to the supply chain
- How and why firms select specific marketing channels
- How elements of the marketing mix (product, price, place and promotion) can be affected.
Contributors and Attributions
- Describing Retailers and Channels. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution
- Apple Logo. Authored by : Apple Inc.. License : Public Domain: No Known Copyright


3.13: Spell Check and Dictionaries
Learning Objectives
- Correct spelling and grammar errors.
As the screenshot above demonstrates, Spell Check and Grammar Check are potentially useful tools, but very fallible ones.
You’ve probably seen the angry red wavy lines indicating a misspelled word as you type a document. Sometimes, those red lines may have been helpful, saving you from making an embarrassing typo. Other times, those red lines are just annoying and distracting. In this section, we will cover how Spellcheck works and how you can best use it to your advantage.
How Spell Checker Works
The spell checker tool compares every word you type against its dictionary, or database of words. If a word isn’t in its dictionary, the word is marked misspelled with a red underline. This means words that are correctly spelled may be marked as misspelled, especially brand names, specialized industry or scientific terms, and words in another language. With a little tweaking, you can teach Microsoft Word (and other Office programs) how to spell uncommon words and names you use regularly. That way, genuine mistakes are not lost in a sea of red underlines.
Practice Question
Adding Words to the Dictionary
The simplest method to add words to the dictionary is to right-click the marked word and select Add. If you do not wish to add the word but do not want to see the red line, you can select Ignore or Ignore All.
Keep in mind that adding a word to the dictionary will not necessarily add all versions of that word. For example, after adding iPad to the dictionary, the dictionary will still not recognize iPads .
The good news is that adding a word to the Microsoft Word dictionary will also add it to the dictionary for other Office products such as Excel and PowerPoint.
The dictionary is case-sensitive. Adding iPad will not add ipad (a good thing, since ipad is incorrect). Keep this in mind if the word you want to add to the dictionary is capitalized because it is at the beginning of a sentence.
Keep in mind that adding a word adds it to whichever language dictionary you are using. For example, adding “c’est la vie” to the dictionary tells Microsoft Word that “c’est la vie” is an English word. That may be nice for the occasional borrowed term, but it will get overwhelming if you find yourself using a lot of foreign words not typically used in English vernacular. We will cover other language dictionaries later in this section.
Bonus: Switching Language Dictionaries
You can switch your dictionary to a different language. Go to Tools > Dictionary, Dictionary, and choose a different language.
Grammar Check
Grammar check works much like spell check, searching through your document for grammatical and style errors rather than spelling errors. Grammar check underlines words or phrases in blue or green, depending on your version of Word.
Right-click on the underlined word or phrase to see suggested corrections. You can also choose Ignore, Ignore Once, or Ignore All if the “error” is actually correct. Like spell check, grammar check is prone to errors. In some versions of Word, you may see an Ignore Rule or “Don’t check for this issue” option. Clicking Ignore Rule will make Grammar check ignore that specific grammar rule in your document while still looking for other grammar errors.
Practice Question
Contributors and Attributions
- Word the critic. Authored by : Anders Sandberg. Located at : https://www.flickr.com/photos/arenamontanus/2753162618 . License : CC BY: Attribution
- Spellcheck and Dictionaries. Authored by : Lumen Learning. License : CC BY: Attribution


5.4: Implement Nutritional Strategies to Impact Neurologic Wellness
By the end of this section, you should be able to:
- 5.3.1 Assess the client for readiness to learn.
- 5.3.2 Teach nutritional strategies to optimize neurologic wellness.
Holistic Nursing Assessment of the Client
When the nurse is considering changes to a client’s diet to facilitate adequate functioning of the neurologic system, it is critical to evaluate how these changes will be affected by the client’s lifestyle and relevant socioeconomic factors. The nurse must also consider the client’s culture, beliefs, attitudes, daily eating patterns, and health conditions that may impact their approach to eating.
Daily Pattern of Food Intake and Sleep
The time of day that meals are taken can improve neurologic health. For example, recent studies suggest that narrowing the window of time during which food is consumed during the day may positively impact cognitive function (Gudden et al., 2021). Intermittent fasting , defined as either eating only between certain hours of the day (e.g., between 11 a.m. and 5 p.m.) or consuming few calories on certain days and eating normally on other days, has been shown to improve neurologic health among persons with Parkinson’s disease, ischemic stroke, mood and anxiety disorders, and epilepsy (Gudden et al., 2021). Maintaining regular hours for meals and sleep has also been associated with positive cognitive performance, and avoiding food intake in the hours before bedtime has been shown to improve cognitive performance and sleep quality (Gudden et al., 2021).
Challenges in Healthy Nutrition
For many Americans, the availability and cost of healthy foods are major concerns. Soaring prices of weekly groceries are concerning, and many Americans rely on food banks or other forms of public assistance to feed their families than ever before. In addition, food preferences and dietary patterns are often influenced by cultural norms, beliefs, family traditions, and long-term habits; therefore, changing one’s dietary intake is often challenging. Dietary preferences are very personal for many people, so making changes may be fraught with high levels of resistance. It is thus important for the nurse to ensure that changes in a client’s diet are feasible and in synergy with the client and their family.
Client Teaching
Given the complexities of dietary habits, it is critical to first understand the client and family’s health state (cognitive state and motivation, swallowing difficulties, mobility concerns), dietary history, allergies, or sensitivities to certain foods, as well as food availability and financial concerns when planning a dietary lifestyle. Client teaching should be part of an interprofessional plan of care involving the nurse, a registered dietitian, and the health care provider. The health care team will be best able to create dietary adaptations that will meet specific criteria that will allow the client to adhere to the plan of care.
Recommended Diet Plans for Comorbid Conditions
For many clients, comorbid conditions can complicate the ability to manifest a comprehensive diet plan. For example, although the Mediterranean diet and ample fluid intake may improve cognitive status, for clients in heart or kidney failure, specific modifications will be required to consider the effect of excess fluid or the correct percentage of protein intake.
Read the following clinical scenario and then answer the questions that follow. This case study is a follow-up to Case Study Part A.
Jamal Powell has a normal PET scan result and is advised to follow up with the clinic regularly. The client is referred to a dietician but misses 6 separate appointments over 2 years.
Jamal is visiting the clinic today for a routine follow-up 2 years after the initial clinic visit. During this time, Jamal’s memory has progressively worsened. Jamal believes these deficits have not been obvious to friends.
The client describes a recent problem of forgetting where they put things and confusing things in their memory. However, the client’s relationship to time, as well as their skills in math and literacy, have not changed. Jamal is independent with regard to basic activities of daily living, but complex tasks are becoming more challenging.
The client states that they have started to feel less confident about driving over the past few months. Jamal appears to be irritable and becomes distressed when discussing the symptoms and recent events they have been dealing with. The client discusses their fears and frustrations surrounding these changes and feels they are becoming depressed about the situation.
The nurse discusses Jamal’s diet and activity and plans to work with the client to improve their nutritional state. Jamal’s diet is high in saturated fats and low in fresh vegetables and fruits. The client’s spouse mentions that they read that polyphenols are good for brain health and asks which foods would be best to include in their diets.
Jamal also asks why everyone keeps saying that sugar in the diet is bad, stating that they have enjoyed sweets their whole life. This is apparent with the recent HbA1c test result, which is 6.3%. (The expected range for nondiabetic clients, such as Jamal, is less than 5.7%.)
The nurse explains that the combination of Jamal’s high BMI of 40, HbA1c of 6.3%, total cholesterol level of 250 mg/dL, and diagnosis of chronic hypertension (today’s blood pressure was 156/98 mm Hg) classifies them as having a condition called metabolic syndrome. The nurse sets up another appointment with the dietitian to help Jamal correct the nutritional issues identified. Because the dietitian’s earliest appointment is 2 months away, the nurse teaches Jamal and their spouse about some basic nutritional changes they can make before the next appointment.
-
Dairy and eggs
-
Fatty fish and oils
-
Colorful fruits and vegetables
-
Peanut butter and almond butter


17.0: Introduction
Good nutrition is essential to healthy living and disease prevention. Nutritious meals provide nutrients to power all the cells of the body and enable biochemical processes that humans need to engage in an active, satisfying life. While nutritional needs across a person’s life span vary, the need for quality, nutrient-rich food never changes. Regardless of their cultural, religious, or economic background, everyone benefits from healthy lifestyle choices and dietary patterns.
Unfortunately, unhealthy dietary patterns not only risk immediate health needs but are also associated with the development of chronic disease. In the United States, 60 percent of the population has been diagnosed with at least one chronic medical condition due to unhealthy dietary practices (USDA, 2020). Whether it is diabetes, heart disease, obesity, or eating disorders, no one is immune from the consequences of poor nutrition. Nurses are on the forefront of health promotion; thus, it is essential for nurses to have a clear understanding of the relationship between food and health. Nurses must teach the importance of good nutrition and understand how specialized diets are used to promote optimal health. Most importantly, nurses must be competent in the skill of nutrition assessment and evaluation across a person’s life span.


27.3: Animal Phylogeny
Skills to Develop
- Interpret the metazoan phylogenetic tree
- Describe the types of data that scientists use to construct and revise animal phylogeny
- List some of the relationships within the modern phylogenetic tree that have been discovered as a result of modern molecular data
Biologists strive to understand the evolutionary history and relationships of members of the animal kingdom, and all of life, for that matter. The study of phylogeny aims to determine the evolutionary relationships between phyla. Currently, most biologists divide the animal kingdom into 35 to 40 phyla. Scientists develop phylogenetic trees, which serve as hypotheses about which species have evolved from which ancestors
Recall that until recently, only morphological characteristics and the fossil record were used to determine phylogenetic relationships among animals. Scientific understanding of the distinctions and hierarchies between anatomical characteristics provided much of this knowledge. Used alone, however, this information can be misleading. Morphological characteristics may evolve multiple times, and independently, through evolutionary history. Analogous characteristics may appear similar between animals, but their underlying evolution may be very different. With the advancement of molecular technologies, modern phylogenetics is now informed by genetic and molecular analyses, in addition to traditional morphological and fossil data. With a growing understanding of genetics, the animal evolutionary tree has changed substantially and continues to change as new DNA and RNA analyses are performed on additional animal species.
Constructing an Animal Phylogenetic Tree
The current understanding of evolutionary relationships between animal, or Metazoa , phyla begins with the distinction between “true” animals with true differentiated tissues, called Eumetazoa , and animal phyla that do not have true differentiated tissues (such as the sponges), called Parazoa . Both Parazoa and Eumetazoa evolved from a common ancestral organism that resembles the modern-day protists called choanoflagellates. These protist cells strongly resemble the sponge choanocyte cells today (Figure \(\PageIndex{1}\)).
Eumetazoa are subdivided into radially symmetrical animals and bilaterally symmetrical animals, and are thus classified into clade Bilateria or Radiata, respectively. As mentioned earlier, the cnidarians and ctenophores are animal phyla with true radial symmetry. All other Eumetazoa are members of the Bilateria clade. The bilaterally symmetrical animals are further divided into deuterostomes (including chordates and echinoderms) and two distinct clades of protostomes (including ecdysozoans and lophotrochozoans) (Figure \(\PageIndex{2}\)). Ecdysozoa includes nematodes and arthropods; they are so named for a commonly found characteristic among the group: exoskeletal molting (termed ecdysis). Lophotrochozoa is named for two structural features, each common to certain phyla within the clade. Some lophotrochozoan phyla are characterized by a larval stage called trochophore larvae, and other phyla are characterized by the presence of a feeding structure called a lophophore.
Link to Learning
Explore an interactive tree of life here. Zoom and click to learn more about the organisms and their evolutionary relationships.
Modern Advances in Phylogenetic Understanding Come from Molecular Analyses
The phylogenetic groupings are continually being debated and refined by evolutionary biologists. Each year, new evidence emerges that further alters the relationships described by a phylogenetic tree diagram.
Link to Learning
Watch the following video to learn how biologists use genetic data to determine relationships among organisms.
Nucleic acid and protein analyses have greatly informed the modern phylogenetic animal tree. These data come from a variety of molecular sources, such as mitochondrial DNA, nuclear DNA, ribosomal RNA (rRNA), and certain cellular proteins. Many evolutionary relationships in the modern tree have only recently been determined due to molecular evidence. For example, a previously classified group of animals called lophophorates, which included brachiopods and bryozoans, were long-thought to be primitive deuterostomes. Extensive molecular analysis using rRNA data found these animals to be protostomes, more closely related to annelids and mollusks. This discovery allowed for the distinction of the protostome clade, the lophotrochozoans. Molecular data have also shed light on some differences within the lophotrochozoan group, and some scientists believe that the phyla Platyhelminthes and Rotifera within this group should actually belong to their own group of protostomes termed Platyzoa.
Molecular research similar to the discoveries that brought about the distinction of the lophotrochozoan clade has also revealed a dramatic rearrangement of the relationships between mollusks, annelids, arthropods, and nematodes, and a new ecdysozoan clade was formed. Due to morphological similarities in their segmented body types, annelids and arthropods were once thought to be closely related. However, molecular evidence has revealed that arthropods are actually more closely related to nematodes, now comprising the ecdysozoan clade, and annelids are more closely related to mollusks, brachiopods, and other phyla in the lophotrochozoan clade. These two clades now make up the protostomes.
Another change to former phylogenetic groupings because of molecular analyses includes the emergence of an entirely new phylum of worm called Acoelomorpha. These acoel flatworms were long thought to belong to the phylum Platyhelminthes because of their similar “flatworm” morphology. However, molecular analyses revealed this to be a false relationship and originally suggested that acoels represented living species of some of the earliest divergent bilaterians. More recent research into the acoelomorphs has called this hypothesis into question and suggested a closer relationship with deuterostomes. The placement of this new phylum remains disputed, but scientists agree that with sufficient molecular data, their true phylogeny will be determined.
Summary
Scientists are interested in the evolutionary history of animals and the evolutionary relationships among them. There are three main sources of data that scientists use to create phylogenetic evolutionary tree diagrams that illustrate such relationships: morphological information (which includes developmental morphologies), fossil record data, and, most recently, molecular data. The details of the modern phylogenetic tree change frequently as new data are gathered, and molecular data has recently contributed to many substantial modifications of the understanding of relationships between animal phyla.
Glossary
- Ecdysozoa
- clade of protostomes that exhibit exoskeletal molting (ecdysis)
- Eumetazoa
- group of animals with true differentiated tissues
- Lophotrochozoa
- clade of protostomes that exhibit a trochophore larvae stage or a lophophore feeding structure
- Metazoa
- group containing all animals
- Parazoa
- group of animals without true differentiated tissues


5.3: Connecting Theories to Environments
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Environments should be planned with developmental theories in mind:
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Jean Piaget and the Cognitive Theory
: Environments should encourage active learning, stimulate skills of inquiry and promote problem-solving/risk-taking. Examples: Materials are placed low on shelves that make easy access for children. The shelves should be labeled with pictures of the materials so that children are encouraged to place these materials where they belong by matching the material to the photo when they finish playing with them. The environment must be stimulating for children to encourage knowledge seeking.
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Vygotsky and the Sociocultural Theory
: Environments should provide opportunities for meaningful interactions that challenge children (zpd) and in which scaffolding exists through child-child and child-adult interactions.
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Behavioral Theory
: Daily routines must be consistent and expectations of behavior should be clearly defined.
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Erikson
: Environments provide opportunities for children to develop feelings of trustworthiness, autonomy and initiative in the early years. The environment provides areas for children to feel safe, play independently, and make choices throughout the day.
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References
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21.2D: Structures Used in Voice Production
Voices produce sounds through a steady flow of air through the larynx, which causes vibrations and creates fluctuations in air pressure.
- Describe the anatomy of voice production structures in the respiratory system
Key Points
- The three basic mechanisms of voice production are air supply, vibration, and resonance.
- Pressure and air-flow speed through the larynx determine the strength and volume of voice.
- The articulation of consonants involves parts of the vocal tract obstructing phonation, and can be active or passive.
- A vowel is any articulation that comes from an open vocal tract.
- Vowel articulation depends mainly on the shape of the lips, position of the tongue, but the shape of the vocal folds are involved as well.
Key Terms
- Resonance : The amplication of vibration by the structures of the upper respiratory tract, which can also influence the quality or tone of the sound.
- Articulation : The process by which raw phonation from the vocal cords is refined into specific sounds, such as consonants and vowels.
- glottis : An organ of speech located in the larynx and consisting of the true vocal cords and the opening between them.
Voice production is a complex process with many different layers and intricacies. The three basic mechanisms of voice production are air supply, vibration, and resonance.
Passive and active articulation shapes and refines phonation (vocal sound production) into the sounds and words used in communication. Voice production is an important physiological process because it enables complex communication between humans.
While the brain is responsible for higher organization and understanding language, the structures of the respiratory system are largely responsible for the production of sound itself.
Basic Mechanisms of Voice Production
Sound is produced by a combination of different structures of the respiratory system working together to create and resonate a sound. There are three basic mechanisms by which the human body produces a voice.
- Air Supply: In order for voice to be produced, air must flow through the vocal folds. The supply of air for phonation comes from the lungs, and the speed and pressure by which is flows through the vocal folds is determined by the diaphragm. The speed of air flow also determines the strength and loudness of the voice.
- Vibration: The vocal folds in the glottis of the larynx vibrate as air passes through them. The vibration creates changes in air pressure that manifest as audible sound waves. They only vibrate if the vocal folds are in the closed position, when the folds are held together by the movement of ayrtenoid cartilage. The pitch of the vibration depends on the length and tension of the vocal folds, which can be altered by muscle action.
- Resonance: The structures of the upper respiratory tract—particularly the soft palate of the mouth, the nasopharynx, and the paranasal sinuses —resonate and amplify the vibration of the vocal folds, making the sound louder and changing its tone. It works similarly to the way the sounding board of a guitar amplifies the vibration of the strings.
These basic mechanisms work together to create the voice. If they are altered, the produced voice will also be altered as well.
For example, during loud voice production, such as shouting or singing, a greater air supply and greater pressure for the flow of air through the vocal folds is required to produce the louder sound. The diaphragm must contract harder to support this greater flow of air compared to normal speech.
Similarly, whispering takes less air compared to normal speech, because the sound produced during whispering is much weaker in comparison.
Articulation
Articulation is the process by which phonation is refined into the specific consonants and vowels used to form words. The articulation of consonants occurs at a point of either active or passive articulation, which is a place in the vocal tract where an obstruction stops the sound.
After the sound is obstructed, the pressure from the air builds based on the shape of that obstruction, which changes the sound into the form it is vocalized as. Vowels are articulated sounds that do not come from obstruction, and instead come from an open vocal tract.
Passive Place of Articulation
The passive place of articulation is the place on the more stationary part of the vocal tract where the articulation occurs. It can be anywhere from the lips, upper teeth, gums, roof of the mouth, or the back of the throat. These areas are passive because no specific action or activity is involved within that area to pronounce the consonant.
Passive articulation is considered a continuum because the obstruction of many different places is needed to produce most of the consonants. There are also several different combinations of areas that can produce the same consonant; for example, many languages may distinguish consonants by articulating them in different areas. Passive places of articulation include:
- The upper lip (labial).
- The upper teeth, either on the edge of the teeth or inner surface (dental).
- The alveolar ridge, the gum line just behind the teeth (alveolar).
- The back of the alveolar ridge (post-alveolar).
- The hard palate on the roof of the mouth (palatal).
- The soft palate further back on the roof of the mouth (velar).
- The uvula hanging down at the entrance to the throat (uvular).
- The throat itself, also known as the pharynx (pharyngeal).
- The epiglottis at the entrance to the windpipe, above the voice box (epiglottal).
Active Place of Articulation
The articulatory gesture of the active place of articulation involves the more mobile part of the vocal tract. This is typically some part of the tongue or lips. It is considered active because these areas change the consonant pronounced by moving or changing.
The active places of articulation are not considered a continuum (unlike passive articulation) because they work independently of each other, but they have the capacity to work together for certain consonants. Active places of articulation include:
- The lower lip (labial).
- Various parts of the front of the tongue.
- The back of the tongue. The aryepiglottic folds at the entrance to the larynx (also epiglottal).
- The glottis (laryngeal).
Vowels
A vowel is a sound that comes from an open vocal tract, and does involve strict obstruction of the sound as with consonants. Therefore, there is more variation in the mechanisms used to create vowels compared to consonants. Vowels are mainly articulated by the shape of the lips, the position of the tongue (both vertical and horizontal), and by the phonation of the larynx itself.
Places of articulation for voice production : Places of articulation (active and passive): 1. Exo-labial (outer part of lip), 2. Endo-labial (inner part of lip), 3. Dental (teeth), 4. Alveolar (front part of alveolar ridge), 5. Post-alveolar (rear part of alveolar ridge & slightly behind it), 6. Pre-palatal (front part of hard palate that arches upward), 7. Palatal (hard palate), 8. Velar (soft palate), 9. Uvular (a.k.a. Post-velar; uvula), 10. Pharyngeal (pharyngeal wall), 11. Glottal (a.k.a. Laryngeal; vocal folds), 12. Epiglottal (epiglottis), 13. Radical (tongue root), 14. Postero-dorsal (back of tongue body), 15. Antero-dorsal (front of tongue body), 16. Laminal (tongue blade), 17. Apical (apex or tongue tip), and 18. Sub-laminal (also known as sub-apical; underside of tongue)


14.4E: Siderophores
- Compare and contrast the role of various siderophores in pathogenecity, including: yersiniabactin, enterobactin and ferrichromes
Siderophores are specific types of molecules utilized by microorganisms to obtain iron from the environment. Specifically, in regards to pathogenicity, organisms that exhibit the ability to produce siderophores release these iron-specific molecules and scavenge iron from their hosts organisms. The siderophores are then utilized by the pathogen to obtain iron. Therefore, siderophores are chelating agents that bind the iron ions. The ability of pathogens to obtain iron from the host is essential for survival because the iron is limited in the host environment, in particular, the host tissues and fluids. The iron is used to allow for formation of soluble ferric ion (Fe3+) complexes that are necessary for maintenance of homeostatic mechanisms within the pathogen.
The ability to form water soluble Fe3+ complexes is a key component to the active transport of the Fe-siderophore complex across the cellular membrane. In iron deficient environments, the siderophores are released and allow for the formation of water soluble-Fe3+ complexes to increase iron acquisition. The complexes then generally bind to the cellular membrane using cell specific receptors. They are transported across the membrane utilized for the necessary processes. However, there are differences in the mechanisms employed by various sideorophobes to obtain iron and the specific type of siderophore utilized varies.
Yersiniabactin
The pathogenic bacteria, Yersinia pestis, Yersinia pseduotuberculosis, and Yersinia enterocolitica have the ability to produce a siderophore called yersiniabactin. Pathogenic yersinia is responsible for numerous diseases including the bubonic plague. The ability of pathogenic Yersinia to establish and spread disease is based on its ability to obtain iron for fundamental cellular processes. In areas of low iron, the organism will release yersiniabactin to form Fe3+ complexes. The yersiniabactin-Fe3+ complex will then bind to the outer membrane of the bacteria based on specific receptor recognition. The complex is then translocated through the membrane via membrane-embedded proteins and iron is released from the yersiniabactin. The iron will then be utilized in numerous cellular processes.
Enterobactin
Pathogenic bacteria such as Escherichia coli and Salmonella typhimurium have the ability to produce a siderophore called enterobactin. This specific type of siderophore is the strongest identified siderophore, to date, with an extremely high binding affinity to Fe3+. Upon a decrease in iron, the bacterial cells release enterobactin which forms a complex with Fe3+. The complex is then transported intracellularly via an ATP-binding cassette transporter. Once the enterobactin-Fe3+ complex arrives intracellularly, it is necessary to remove the Fe3+ from the complex. Due to the high-binding affinity of enterobactin, the bacteria require a highly specific enzyme, ferrienterobactin esterase, to cleave the iron from the complex. The iron released from the complex will then be utilized in metabolic processes.
Ferrichrome
Another type of siderophore produced by pathogenic fungi includes a ferrichrome. Fungi that have been shown to produce ferrichromes include those in the genera Aspergillus, Ustilago, and Penicillum. The ferrichrome allows for formation of a ferrichrome-iron complex which can then interact with a protein receptor on the cell surface. The ferrichrome promotes iron transport within the organism to allow metabolic processes to occur.
The discovery and identification of siderophores have allowed for the development of treatments targeting these siderophore-iron complexes. By targeting these complexes, the pathogenic microorganisms can be targeted by inhibiting necessary cellular processes. The production and importance of these siderophores to pathogenic organisms is key to their survival.
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Key Points
- Siderophore – iron complexes are necessary for iron acquisition to various pathogenic organisms for metabolic processes.
- The types of siderophores produced are species specific and exhibit different properties.
- The siderophores are necessary to obtain iron by binding to cell surfaces and transporting the siderophore-iron complexes intracellularly.
- Siderophores are produced in environments that have low iron concentration, such as host tissues and fluids. They are considered advantageous to pathogenic organisms.
Key Terms
- siderophore : Any medium-sized molecule that has a high specificity for binding or chelating iron; they are employed by microorganisms to obtain iron from the environment
- chelating agent : A compound that reacts with a metal ion to produce a chelate.