15.6: Editing Focus- Words Often Confused
By the end of this section, you will be able to:
- Define and give examples of homonyms, homographs, and homophones.
- Demonstrate correct use of commonly confused words.
When writing about language, you don’t want to make the mistake of misusing language. Homonyms are words that sound the same and are spelled the same but have different meanings. Homographs are words that are spelled the same but may sound different and have different meanings. Homophones are words that sound the same but might have a different spelling, a different meaning, or both. Because they sound the same, they are often confused. Other words frequently confused sound slightly different and have slightly different spellings. It’s important to recognize and correct commonly confused words because spell checkers don’t always catch these errors.
Homonyms
Because homonyms’ spelling and pronunciation are the same but their meanings are different, you probably don’t have to worry about misusing them, although they often belong to different parts of speech. However, some awareness of homonyms and their multiple meanings will help ensure that your language is precise. For more information about homonyms, see “Words and Language.”
Here are some common homonyms and their meanings:
| band |
Verb: to come together Noun: group of musicians |
| rock |
Verb: to move back and forth Noun: hard material found in the earth |
| tire |
Verb: to become sleepy or lose energy Noun: ring-shaped rubber component that fits over the wheel of a vehicle |
| current |
Noun: flow of water or electricity Adjective: up to date |
| train |
Verb: to teach or learn through practice Noun: group of connected railroad cars |
Homographs
Homographs have the same spelling but different meanings. They are considered different from homonyms because they may be pronounced differently. Some disagreement exists among scholars as to whether pronunciation must be different for a word to be considered a homograph. Some experts insist that origins must differ as well. However, all agree that meanings differ. For more information about homographs, see Words and Language.
Here are some common homographs and their meanings:
| lead |
Noun: a metal (pronounced led ) Verb: to be at the front, to guide (pronounced leed ) |
| bow |
Verb: to bend forward Noun: a position in which one bends forward Noun: the front of a ship Noun: a type of knot (pronounced boh ) |
| object |
Noun: a material item; a grammatical term (accent on first syllable) Verb: to express disapproval (accent on second syllable) |
| close |
Adjective: near ( s pronounced like ss ) Verb: to bar passage; to shut down ( s pronounced like z ) |
| minute |
Noun: 60 seconds (pronounced minit ) Adjective: tiny (pronounced my-noot or my-nyoot ) |
Homophones
Homophones are words that are spelled differently and have different meanings but sound the same. Homophones are often the words that cause the most confusion and the most frequent errors. For more information about homophones, see Words and Language.
Here are some common homophones and their meanings:
| weather/ whether |
Weather means the conditions outside, such as temperature and sunshine. Whether is a conjunction that joins words or other parts of a sentence. |
| meat/meet |
Meat is animal-based food. When you meet someone, you encounter or get together with that person. |
| hear/here |
Hear is what you do when you listen to music. Here is a place nearby. |
| two/too/to |
Two is a number, more than one and less than three. Too may mean “also,” or it may mean “very” or “more (adjective) than desired.” To is a preposition indicating direction or motion, as in “I wrote to the manager after I’d gone to the office.” |
The following homophones are often confused. Note that one of each group is a contraction : two words shortened and joined by an apostrophe.
| their | Possessive pronoun, belonging to them : “The family packed their suitcases for their vacation.” Both the suitcases and the vacation belong to the family . |
| they're | Contraction joining they and are : “ They’re leaving for the airport at 10:30.” In other words, they are leaving. |
| there | Referring to a location: “‘Baggage claim is over there ,’ the airport worker said while pointing to the baggage carousels.” This form indicates a physical place. |
| your | Possessive pronoun, belonging to you : “ Your new phone will be mailed on November 4.” The phone belongs to you . |
| you're | Contraction joining you and are : “ You’re registered for the conference.” In other words, you are registered. |
| its | Possessive pronoun, belonging to it : “ Its positive qualities outweigh its negative ones.” Both the positive and the negative qualities belong to it . It could be a book, house, car, movie, phone, or any other object or idea. |
| it's | Contraction joining it and is : “ It’s raining today.” In other words, it is raining. |
What is the Effect ? Or Is It Affect ?
One final category of commonly confused words are words that sound and look similar but have different meanings.
| Affect |
Verb: to influence an outcome: “Increasing the minimum wage will affect the incomes of millions of Americans.” Noun: body language that accompanies an expression of emotion: “The crime victim showed normal reactions and affects .” |
| Effect |
Noun: the outcome or result of an influence: “The effect of the increased minimum wage will be a 10 percent decrease in the federal poverty rate.” Verb: to cause to come into existence: “Congress will effect the new law.” |
| Insure | Verb: to protect against damage or loss: “Homeowners must insure their property for the cost of full replacement." |
| Ensure | Verb: to make certain of: “This policy will ensure that the bank doesn’t incur a loss as a result of irreparable damage to the home.” |
Practice
Choose the correct word to complete each sentence.
- Can you (hear/here) the phone ring from (hear/here)?
- Are you going (to/too/two) the movie theater at (to/too/two) p.m., or is that (to/too/two) early for you?
- This show always has a depressing (affect/effect) on me.
- (Its/It’s) hard to find small apartments for rent in this neighborhood.
- Let’s (meet/meat) at the pizza place so that I can eat something without (meet/meat).
- Although it will need work, (its/it’s) basic structure is sound.
- Check about (ensuring/insuring) personal items in your apartment.
- You can set the groceries on the table over (their/there/they’re).
- I’m going to watch the game at (their/there/they’re) house.
- How do you think the elections will (affect/effect) the neighborhood?
- (Their/There/They’re) listening to music on (their/there/they’re) headphones.
- (You’re/Your) going to trip if you don’t tie (you’re/your) shoelaces.
- Bring (you’re/your) phone to the store to exchange it for a new one.
- (Weather/whether) we travel depends on the (weather/whether).


7.2.26: Image Choice (Usable with Workarounds)
Query \(\PageIndex{1}\)
Overall Usability
Usable with Workarounds:
- The activity requires significant adjustments by content authors.
Important Reminders
Any text added in the H5P Question text form field by content authors is part of the information announced by screen reading software. However, H5P allows content authors to include content and formatting that screen reading software is not able to announce as part of the accessible label for the question. H5P does not provide an option to include a separate question label (such as a correctly structured heading) or separate instructions for questions.
Use best practices for content authoring, including:
- Begin the activity content with a correctly structured heading
- Add instructions for questions on the page outside of the H5P content
- Provide users with instructions that describe the Show solution and Retry button icons for users who require magnification.
Users of screen reading software may need to be provided with an alternative activity to be able to review correct and incorrect answers.
Visible Indication of Focus
The activity buttons and interactive elements have a visible indication of focus.
Color Contrast
The visible indication of focus has good contrast. The color contrast of meaningful visual elements and text with backgrounds must be ensured manually by the content author.
Keyboard Usability
The activity is usable with the keyboard.
No Keyboard Traps
There are no keyboard traps.
Magnification
The activity supports both magnification up to 200% and reflow of content with magnification up to 400% without a loss of content or functionality and without requiring horizontal scrolling. The Retry button text is not visible at 200%, the Show solution and Retry button text is not visible at 200% or 400%, and the tooltips for the buttons do not magnify, but the icons do.
Usability with Screen Reading Software
The activity is not usable with screen reading software. The correct and incorrect answers cannot be announced by screen reading software.


16.3: External and Internal Monitoring
By the end of this section, you will be able to:
- Explain how to perform intermittent auscultation of the fetal heart rate and manual palpation of uterine contractions, as well as state the advantages and disadvantages of intermittent auscultation
- Explain the placement, advantages and disadvantages, and indications for continuous external electronic fetal heart rate and uterine contraction monitoring
- Interpret the electronic fetal heart rate and uterine contraction monitor graph
- Document the assessment data of the electronic fetal heart rate and uterine contraction monitor graph
Monitoring the pregnant person and the fetus and analyzing the FHR and contraction patterns are essential parts of the nurse’s role during the antepartum period and the labor and birth process, allowing the health-care team to anticipate and prevent complications. The nurse educates the pregnant person regarding what type of monitoring is recommended at certain stages of pregnancy and labor. Providing a choice of monitor type when applicable encourages shared decision making during this important time for the person and the family.
Three methods of monitoring the fetal heart rate and uterine contractions can be used when the nurse is caring for the antepartum or intrapartum pregnant person. The first is intermittent auscultation; the other two methods—one performed by attaching external electronic monitors and the other by inserting internal electronic monitors—provide the health-care team with continuous feedback on the fetal heart rate and uterine contractions.
Intermittent Auscultation
The technique of assessing fetal well-being by listening to and counting the fetal heart rate for a specified amount of time and at specified intervals, depending upon the stage of labor, is called intermittent auscultation (IA) . The nurse palpates the uterus to evaluate the contractions’ duration (from the beginning to the end of the contraction), intensity, and frequency (from the beginning of one contraction to the beginning of the next contraction). The nurse listens before, during, and after the contraction to evaluate for any FHR changes. The nurse may perform IA with either a fetoscope or a Doppler monitor. The fetoscope is similar to a stethoscope but also uses bone conduction to allow for better detection of the FHR. The fetoscope can be used after 20 weeks’ gestation; prior to that time, the FHR is not detectable by fetoscope. A Doppler monitor is a handheld device that uses ultrasonic waves to detect the FHR. Dopplers are used in antepartum visits as well as during labor. When auscultating the FHR, it is important to assess the maternal heart rate at the same time to ensure the Doppler is providing fetal data.
Determining Optimal Placement of the Auscultation Device Using Leopold’s Maneuvers
Determining the position of the fetus is important when placing the ultrasound transducer or fetoscope on the abdomen of the laboring person. Leopold’s maneuvers are performed to assist in identifying the fetal position. The nurse performs the maneuvers, locates the fetal back, and then places the transducer on the laboring person’s abdomen directly over the fetal back where the fetal heart is the loudest. Figure 16.16 illustrates where to hear the FHR most clearly when the fetus is vertex or breech. Figure 16.17 illustrates Leopold’s maneuvers to determine the fetal presentation and position.
The nurse explains to the patient that these maneuvers require touching of the abdomen to discover how the fetus is lying inside the uterus. The nurse will lower the head of the bed with the patient on their back and begin the assessment.
- The first maneuver is done facing the patient. Both hands palpate the top of the abdomen to determine the shape, mobility, and size of the fetal part. A hard, round fetal part is the head and would diagnose the fetal position as noncephalic. A larger, softer fetal part is the buttock or back of the fetus.
- The second maneuver is used to determine the placement of the fetal back. After determining what is in the fundus, the hands slide down to the sides of the uterus and determine the fetal lie. If the nurse determines the fetal lie is longitudinal (or oblique), using the palms, the right hand holds the uterus while the left palpates for a smooth long back or small, “bumpy” extremities. The opposite side is then palpated. If the nurse determines the fetal lie is transverse, the nurse will palpate the fetal head on one side and lower extremities on the opposite side. The fetal back will face either the fundus or the cervix.
- The third maneuver determines the presenting part in the pelvic inlet. Holding the hand in a “C,” the thumb and fingers grasp the fetal part in the low abdomen above the pubic bone. Again, if the part is hard and round, it is the fetal head. The nurse can attempt to move the head back and forth to determine if the head is engaged in the pelvis.
- The fourth and last maneuver is used to determine the fetal descent into the pelvis. Turning to face the laboring person’s feet, both hands palpate down the sides of the uterus to the pubic bone and presenting part. The brow cannot be palpated if the fetal head is well flexed and engaged.
After performing the Leopold maneuvers and determining the placement of the fetal presentation and back, the nurse places the ultrasound transducer or Doppler to assess FHR.
Determining Appropriate Intervals for Intermittent Auscultation
Timing of IA is determined by the stage of labor, facility protocols, and health-care provider’s orders. AWHONN (2018) has published recommended intervals for IA based on stages of labor. Table 16.3 summarizes these guidelines for IA. The FHR is auscultated before, during, and after a contraction.
| Latent Phase Labor (4–5 cm) | Active Phase Labor (≥6 cm) | Second Stage Labor Passive Descent | Second Stage Active Pushing |
|---|---|---|---|
| Every 15–30 minutes | Every 15–30 minutes | Every 15 minutes | Every 5–15 minutes |
With intermittent auscultation , the nurse will also monitor uterine contractions by palpation. The nurse’s hand is placed at the top of the fundus prior to a contraction to determine baseline uterine tone. As the uterus contracts, the nurse feels a hardening of the uterus. The strength of the contraction is based on the laboring person’s description of the contraction and the nurse’s palpation. A mild contraction is similar to pressing on one’s cheek; a moderate contraction is similar to pressing on the tip of the nose; and a strong contraction is similar to pressing on the forehead. The nurse also notes the frequency and duration of the contraction during palpation.
Advantages and Disadvantages of Intermittent Auscultation
One advantage of IA is the freedom of movement it allows the laboring person. Movement in labor aids the fetus in its descent into the pelvis and in pain control for the laboring person. Because IA does not require attached monitors, the laboring person can walk, shower, sit on a birthing ball, or take a bath. The lack of monitors strapped onto the abdomen helps the laboring person rest easier between contractions.
One of the disadvantages of IA is the requirement for one-to-one nurse–laboring person staffing. This can be a challenge when the labor and delivery unit is busy and staffing is not available for one-to-one care. Another disadvantage is the lack of monitor tracings. Fetal monitor strips provide a visual history of the uterine contractions and FHR . These strips are used in communication with other health-care providers and provide a permanent record of labor.
Continuous External Monitoring of FHR and Contractions
The most common method of monitoring the fetal heart rate and contraction pattern is through continuous external monitoring . Doing so requires the use of two monitoring devices: an ultrasound device to evaluate the fetal heart rate, and a tocodynamometer (toco) to detect the frequency of uterine contractions. The nurse applies these external monitors to the patient’s abdomen; the devices are held in place with belts or adhesives. Figure 16.18 shows the continuous fetal monitor and the wireless fetal monitoring system. Wireless monitors allow patients to ambulate in labor. Other monitors are waterproof and allow for continuous monitoring during hydrotherapy and water birth.
Both transducers are connected to the monitor, and their signals are traced onto graph paper. The nurse explains to the patient that these monitors are external and can lose the signal with movement of the laboring person or fetus. The nurse explains that the laboring person should not be alarmed if they no longer see the signal on the monitor. They are instructed to call the nurse in these instances so that the monitor can be adjusted.
Through continuous external monitoring, fetal heart rate baseline and variability can be determined, as can periodic fetal heart rate changes. Situations requiring continuous fetal monitoring include meconium-stained fluid, multiple gestation, preeclampsia , epidural or spinal anesthesia, and dysfunctional labor (Simkin et al., 2017).
Application of the External Monitors
Uterine and FHR monitoring can be achieved externally and internally. After performing Leopold’s maneuvers , the nurse will place the external transducer and tocodynamometer on the person’s abdomen. The external monitors provide an FHR tracing.
Applying the Ultrasound Transducer
The ultrasound transducer is placed over the fetal back closest to the fetal head, where the FHR is usually the loudest. The nurse identifies this area using Leopold’s maneuvers as previously described. The transducer is not placed over the fetal chest because movement of the extremities interferes with continuous detection of the FHR. Ultrasound gel is placed on the transducer to allow a medium for ultrasound waves to travel into and out of the uterus. The transducer is then secured by a belt.
Placement of Uterine Tocodynamometer
The toco transducer senses change in the uterine muscle during a contraction. The toco transducer is placed on the abdomen where the fundus is firmest, most often over the fetal buttocks in a cephalic presentation. While performing Leopold’s maneuvers, the nurse can assess for the optimal placement of the toco. Having determined the optimal location, the nurse secures the toco transducer to the laboring person by a belt (Figure 16.19).
Advantages and Disadvantages of External Monitoring
The advantages of external fetal monitoring are its ease of use, the ability of the devices to record and digitally store the monitor results, and the ability for more advanced fetal assessment. This method also does not require one-on-one nurse staffing because the monitors are electronically connected to central monitors, which allow the nurse to read the fetal monitor tracing from any labor room or the nurses’ station and to monitor more than one patient simultaneously. Central monitoring also allows the nursing staff and health-care providers to assess the fetal monitor tracing from all rooms on the unit, from an office, and from home. Additionally, external monitors can be used regardless of the status of the amniotic membrane or the extent of cervical dilation.
A disadvantage of continuous external monitoring is that the required positioning of the monitor can result in the signal being lost when the person changes positions or gets out of bed. If wireless monitoring is not available, continuous monitoring can limit the pregnant person’s freedom to walk, shower, or bathe. Another disadvantage is that the tocodynamometer can determine only the frequency and duration of the contractions, but not the strength of the contractions, which will need to be palpated. As discussed previously, the nurse must palpate the uterus to evaluate the uterine contraction.
Continuous Internal Monitoring
Internal fetal monitors consist of the fetal scalp electrode (FSE) that affixes to the scalp of the fetus and provides close monitoring of the fetal heart rate and an intrauterine pressure catheter (IUPC) that lie inside the uterus and measure the frequency, duration, and strength of the uterine contractions and the resting tone of the uterus. The amniotic sac must be ruptured prior to placement of internal monitors. Nurses and health-care providers use internal monitors in situations that require more in-depth fetal surveillance, such as nonreassuring fetal heart rates , low amniotic fluid, and difficulty in maintaining a continuous FHR tracing, and during the use of uterotonics for labor induction or augmentation.
Application of Internal Monitors
In continuous internal monitoring , an FSE is attached to the fetal scalp, avoiding the fontanelles, and the IUPC is placed into the uterus between the fetus and uterine wall. Figure 16.20 illustrates the internal fetal monitoring system.
Internal monitors are placed by trained health-care providers and labor and birth nurses. A cervical exam is performed for assessment of dilation and membrane status. The cervix must be dilated, and the amniotic membrane must be ruptured to pass the monitors into the uterus. If the nurse assesses a need for internal monitoring, the nurse will notify the health-care provider. If the amniotic sac is intact, the health-care provider must determine the need for the monitors and weigh the pros and cons of rupturing the amniotic membrane. When appropriate, the health-care provider breaks the bag of amniotic fluid using an amniotomy hook , which is a plastic wand with a small hook that snags the membranes. Figure 16.21 illustrates use of the amniotomy hook.
Fetal Scalp Electrode
To place the fetal scalp electrode (FSE) , the fetal head must be low enough to be palpated during the vaginal exam, and the cervix must be open enough to insert the FSE. The head is assessed for suture lines and fontanelles, as seen in Figure 16.22. Avoiding the fontanelles, the nurse or the health-care provider places the scalp electrode firmly against the fetal scalp and turns it clockwise to adhere the electrode. The nurse then connects the scalp electrode transducer to the monitor, and the FHR electrocardiogram is detected and documented in the monitor tracing.
Intrauterine Pressure Catheter
The IUPC, seen in Figure 16.23, is inserted using a firm plastic introducer containing the pressure catheter. The catheter has markers to guide the depth of placement to avoid uterine perforation. To avoid the IUPC being inserted through the placenta, the introducer must be retracted immediately if frank red blood is noted. The pressure catheter transducer is attached to the monitor and traces, in exact millimeters of mercury ( mm Hg ), the strength of the contractions and the uterine resting tone.
Advantages and Disadvantages of Internal Monitoring
Internal monitors are an advantage when conditions are detected that require comprehensive and continuous evaluation of fetal well-being and uterine contractions. The fetal scalp electrode is the most precise way to monitor the fetal heart rate because it receives signals directly from the fetus and transmits them to the monitor tracing. In situations of FHR decelerations , the internal monitor can provide exact timing of the FHR deceleration in relation to the strength of contraction. The scalp electrode also allows the nurse to evaluate variability more precisely. This information is helpful in determining nursing interventions to resolve FHR decelerations. This monitor is also helpful when the FHR is difficult to detect with external monitors. In the situation where labor is not progressing as predicted, the IUPC can detect the strength of contractions. This information can assist the nurse in determining how to manage the oxytocin during labor induction or augmentation. Another advantage is that the health-care provider may use the FSE to rupture the membranes.
The disadvantages of internal monitors are increased risk of infection due to the rupture of membranes and the break in skin integrity necessary to place the fetal scalp electrode. Additionally, internal monitoring introduces the risk of injury to the fetus from the scalp electrode and the IUPC. During placement of the IUPC, there is a risk of puncturing the uterus and placenta, which is why the IUPC is inserted by the health-care provider or by registered nurses with advanced training in some labor and birth units.
Efficacy and Outcomes Data: Use of Continuous Electronic Fetal Monitoring
Continuous electronic fetal monitoring was developed in the 1970s. The original purpose of continuous electronic fetal monitoring was to decrease complications such as newborn seizures, cerebral palsy, and intrapartum fetal demise. According to the Cochrane Library (Alfirevic et al., 2017), continuous FHR monitoring with fetal scalp electrode , when compared to IA, showed no significant improvement in overall rate of fetal deaths but did significantly reduce the rate of neonatal seizures. The research found that intermittent versus continuous fetal scalp electrode monitoring did not increase instrumental or cesarean births, but no difference was seen in the incidence of cerebral palsy. Table 16.4 summarizes the advantages and disadvantages of the different fetal monitoring types.
| Monitor Type | Advantages | Disadvantages |
|---|---|---|
| Intermittent auscultation (IA) |
Freedom of mobility
Ability to shower or bathe Not dependent upon amniotic membrane status |
One-to-one nursing–pregnant person care
Lack of permanent record of FHR and UC |
| Continuous external |
Ease of use
Ability to record and store monitor results Allows the electronic fetal monitoring (EFM) to be read on any computer Not dependent upon amniotic membrane status |
Signal can be lost with movement of the fetus or person
Limits freedom of movement or hydrotherapy Toco does not provide strength of UC |
| Continuous internal |
More precise measurements of FHR and UC
Does not lose signal with movement of the fetus or person IUPC monitor results can help guide use of oxytocin |
Amniotic membrane must be ruptured
Increased risk for infection Risk of injury to the fetus or uterus from the FSE or IUPC devices |
Interpretation of the Electronic Fetal and Uterine Monitor Tracing
Interpretation of the fetal heart rate consists of evaluating the baseline, variability, and periodic changes of the FHR in relation to the uterine contractions. The nurse interprets uterine contractions by evaluating frequency, duration, strength, and resting tone. The contraction frequency is how often the contractions occur and is determined by measuring from the beginning of one contraction to the beginning of the next contraction. The contraction length is how long the contraction lasts and is determined by measuring the time between the beginning of the contraction and the end of the contraction. The contraction strength is the intensity of the contraction and is determined externally by palpation (see earlier discussion of intermittent auscultation ) or internally by the IUPC. Contraction strength cannot be measured accurately when the external toco is used; therefore, the nurse palpates and describes contractions as mild, moderate, or strong. When IUPC is in use, contractions will be described in mm Hg.
Interpreting fetal monitor tracings is a skill that takes practice to refine. This fetal monitoring game reviews fetal monitoring terminology and tracings. The game can be played unlimited times.
Expected uterine activity is defined as five contractions or fewer in a 10-minute period. Expected contraction length is between 45 and 60 seconds and may increase to 90 seconds closer to the onset of the second stage of labor. Abnormal uterine activity called uterine tachysystole is more than five contractions in 10 minutes or a contraction length of 2 or more minutes within a 30-minute period. Uterine tachysystole can be diagnosed in both spontaneous and induced labor.
Efficiency of the uterine contractions during labor can be measured in Montevideo units . The nurse calculates the Montevideo units by subtracting the uterine resting tone in mm Hg from the mm Hg measuring the peak of the contraction for each contraction occurring within a 10-minute time interval (Kissler et al., 2020). Montevideo units can be calculated only when an IUPC is in place.
Fetal well-being is influenced by uterine activity. Therefore, when the nurse is evaluating fetal well-being, both the fetal heart rate and uterine activity, as well as the fetal response to the uterine activity, must be described. A normal fetal monitor tracing is found in Figure 16.24.
Interpretation of fetal heart rate was historically a problem because of inconsistencies in terminology and reader error. In 2008, the National Institute of Child Health and Human Development Working Group created a three-tiered classification system to interpret fetal heart rate patterns (ACOG, 2009). The three-tiered system allows for consistent terminology and management by the health-care provider and nurse. A summary of the three categories is found in Figure 16.25.
Category I tracings are considered normal and do not require further actions by the nurse other than routine nursing care for the antepartum or laboring patient. Figure 16.26 shows a Category I tracing. Category II tracing characteristics do not meet the criteria of either Category I or III. Nursing care requires continued surveillance with consideration of clinical circumstances and reevaluation and notification of the health-care provider. Figure 16.27 shows a Category II tracing. Category III tracings are always considered abnormal and require the nurse to contact the health-care provider with a report of the FHR abnormalities. Nursing interventions aimed at resolving the cause of the abnormality are discussed in 16.5 Intrauterine Resuscitation in this chapter. If interventions do not resolve the abnormality, the birth must be expedited, most commonly via cesarean section (ACOG, 2009). Figure 16.28 shows a Category III tracing.
Documentation of the FHR and Uterine Contraction Pattern
Electronic health record ( EHR ) documentation during labor should include assessment of the pregnant person and the fetal heart rate. Continuous monitors record the fetal heart rate, and the tracings allow the nurse and health-care providers to review the current and previous tracing. The stored tracing also allows the nurse to care for the laboring patient during delivery and review the monitor tracing when needed.
Documentation should show evaluation of the fetal heart rate and contraction pattern every 15 to 30 minutes if there is continuous monitoring, with a summary of fetal status. The documentation should also discuss the frequency of assessment and the interpretation of the fetal heart rate. When labor-inducing medications are in use, such as oxytocin (Pitocin) , the nurse should document fetal status before and after increasing the dose (Simpson, 2014).
Documentation in labor is often determined by hospital policies. Every hospital should have a policy on the frequency of fetal heart rate and contraction pattern interpretation. To ensure all components of the tracing are documented, nurses use a standardized way of describing the FHR tracing. Figure 16.29 is an example of how to organize necessary information for documentation.
A process for evaluating and documenting fetal monitor tracings typically includes the following steps:
- Identify external or internal monitoring.
- Describe FHR baseline: rate and variability.
- Describe the presence or absence of FHR accelerations or decelerations.
- Identify the type of deceleration if present.
- Document the uterine contraction pattern with frequency, duration, intensity, and presence or absence of uterine resting tone.
- Interpret the tracing and identify the tracing as Category I, II, or III.
- Document the plan of care. Communication with the pregnant person and family is documented as part of the EHR . Nurses also document communication with the health-care provider and any new orders received from the provider.
Management of Uterine Tachysystole
Uterine tachysystole is defined as more than 5 contractions in 10 minutes averaged over 30 minutes. When administering oxytocin for induction or augmentation of labor, uterine tachysystole is a risk factor. The following interventions should be used when uterine tachysystole is diagnosed.
For uterine tachysystole with nonreassuring fetal heart rate pattern:
- Discontinue oxytocin.
- Turn laboring patient to lateral position.
- IV bolus with 500 mL lactated Ringer’s (unless contraindicated).
- Apply O 2 via mask at 8 to 10 liters per minute.
- Administer 0.25 mg terbutaline SQ.
For uterine tachysystole with reassuring fetal heart rate pattern:
- Reposition laboring patient (left or right lateral position).
- IV bolus with 500 mL lactated Ringer’s (unless fluid restricted).
- If uterine activity has not returned to normal after 10 minutes, decrease oxytocin rate by half; if uterine activity has not returned to normal after 10 minutes, discontinue oxytocin until uterine activity has returned to normal (≤ 5 contractions in 10 minutes).
- Notify the health-care provider.
- Have terbutaline 0.25 mg SQ readily available.
- If the tachysystole has resolved within 30 minutes, the oxytocin infusion may be restarted at ½ of the previous rate. If the tachysystole has resolved in > 30 minutes, the oxytocin infusion may be restarted at the previous rate.
(Lyndon & Wisner, 2021)


3.1D: Magnification and Resolution
- Define magnification and resolution
Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called “magnification. ” The term magnification is often confused with the term “resolution,” which describes the ability of an imaging system to show detail in the object that is being imaged. While high magnification without high resolution may make very small microbes visible, it will not allow the observer to distinguish between microbes or sub-cellular parts of a microbe. In reality, therefore, microbiologists depend more on resolution, as they want to be able to determine differences between microbes or parts of microbes. However, to be able to distinguish between two objects under a microscope, a viewer must first magnify to a point at which resolution becomes relevant.
Resolution depends on the distance between two distinguishable radiating points. A microscopic imaging system may have many individual components, including a lens and recording and display components. Each of these contributes to the optical resolution of the system, as will the environment in which the imaging is performed. Real optical systems are complex, and practical difficulties often increase the distance between distinguishable point sources.
At very high magnifications with transmitted light, point objects are seen as fuzzy discs surrounded by diffraction rings. These are called Airy disks. The resolving power of a microscope is taken as the ability to distinguish between two closely spaced Airy disks (or, in other words, the ability of the microscope to distinctly reveal adjacent structural detail). It is this effect of diffraction that limits a microscope’s ability to resolve fine details. The extent and magnitude of the diffraction patterns are affected by the wavelength of light (λ), the refractive materials used to manufacture the objective lens, and the numerical aperture (NA) of the objective lens. There is therefore a finite limit beyond which it is impossible to resolve separate points in the objective field. This is known as the diffraction limit.
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- Chlorophyta. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Chlorophyta. License : CC BY-SA: Attribution-ShareAlike
- macroscopic. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/macroscopic. License : CC BY-SA: Attribution-ShareAlike
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- Microscopic. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Microscopic. License : CC BY-SA: Attribution-ShareAlike
- Microscopic. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Microscopic. License : CC BY-SA: Attribution-ShareAlike
- Bacterial growth. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Bacterial_growth. License : CC BY-SA: Attribution-ShareAlike
- genome. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/genome. License : CC BY-SA: Attribution-ShareAlike
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- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//biology/definition/concave. License : CC BY-SA: Attribution-ShareAlike
- specular. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/specular. License : CC BY-SA: Attribution-ShareAlike
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- Optical resolution. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Optical_resolution. License : CC BY-SA: Attribution-ShareAlike
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- diffraction. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/diffraction. License : CC BY-SA: Attribution-ShareAlike
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Key Points
- Magnification is the ability to make small objects seem larger, such as making a microscopic organism visible.
- Resolution is the ability to distinguish two objects from each other.
- Light microscopy has limits to both its resolution and its magnification.
Key Terms
- airy disks : In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best-focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light.
- diffraction : the breaking up of an electromagnetic wave as it passes a geometric structure (e.g., a slit), followed by reconstruction of the wave by interference


2.3.1: Electronegativity and types of Chemical Bonds
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Electronegativity
The electronegativity of an atom is a measure of its affinity for electrons. The atoms of the various elements differ in their affinity for electrons.
This image distorts the conventional periodic table of the elements so that the greater the electronegativity of an atom, the higher its position in the table. Although fluorine (F) is the most electronegative element, it is the electronegativity of runner-up oxygen (O) that is exploited by life. The shuttling of electrons between carbon (C) and oxygen (O) atoms powers life.
- Moving electrons against the gradient (O to C) — as occurs in photosynthesis — requires energy (and stores it).
- Moving electrons down the gradient (C to O) — as occurs in cellular respiration — releases energy.
The relative electronegativity of two interacting atoms also plays a major part in determining what kind of chemical bond forms between them.
Chemical Bonds
Three main types of chemical bonds:Ionic Bond, Covalent Bond, Polar Covalent Bond.
Ionic Bond
Example of an ionic bond is : Sodium (Na) and Chlorine (Cl) = Ionic Bond. There is a large difference in electronegativity between Na and Cl atoms, so
- the chlorine atom takes an electron from the sodium atom
- converting the atoms into ions (Na + ) and (Cl − )
- These are held together by their opposite electrical charge forming ionic bonds
- Each sodium ion is held by 6 chloride ions while each chloride ion is, in turn, held by 6 sodium ions
- Result: a crystal lattice (not molecules) of common table salt (NaCl)
Covalent Bond
Example of a covalent bond is: Carbon (C) and Hydrogen (H) = Covalent Bond. There is only a small difference in electronegativity between the C and H atoms, so
- the two atoms share the electrons
- Result: a covalent bond (depicted as C : H or C - H)
- The atoms are held together by their mutual affinity for their shared electrons
- An array of atoms held together by covalent bonds forms a true molecule
Polar Covalent Bond
Example of a polar covalent bond is: Hydrogen (H) and Oxygen (O) = Polar Covalent Bond. There is a moderate difference in electronegativity, causing the oxygen atom to pull the electron of the hydrogen atom closer to itself. This results in a polar covalent bond. Oxygen does this with 2 hydrogen atoms to form a molecule of water
Molecules, like water, with polar covalent bonds are themselves polar; that is, have partial electrical charges across the molecule and may be attracted to each other (as occurs with water molecules). These species are good solvents for polar and/or hydrophilic compounds may form hydrogen bonds.


6.3: Dynamics of Rotational Motion- Rotational Inertia
Learning Objectives
- Understand the relationship between torque, rotational inertia and angular acceleration.
- Study the turning effect of torque.
- Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration.
If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity as seen in Figure \(\PageIndex{1}\). In fact, your intuition is reliable in predicting many of the factors that are involved. For example, we know that a door opens slowly if we push too close to its hinges. Furthermore, we know that the more massive the door, the more slowly it opens. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration; another implication is that angular acceleration is inversely proportional to mass. These relationships should seem very similar to the familiar relationships among force, mass, and acceleration embodied in Newton’s second law of motion. There are, in fact, precise rotational analogs to both force and mass.
To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force \(F\) on a point mass \(m\) that is at a distance \(r\) from a pivot point, as shown in Figure \(\PageIndex{2}\). Because the force is perpendicular to \(r\), an acceleration \(a=\frac{F}{m}\) is obtained in the direction of \(F\). We can rearrange this equation such that \(F=m a\) and then look for ways to relate this expression to expressions for rotational quantities. We note that \(a=r \alpha\), and we substitute this expression into \(F=m a\), yielding
\[F=m r \alpha . \nonumber \]
The turning effectiveness of force is called torque . Intuitively, torque increases with a larger lever arm , or the perpendicular distance between the center of rotation and the point at which force is being applied (think of a see-saw; to balance the weight of a heavier child on one side with a lighter child on the other side, the lighter child needs to sit farther from the pivot point). So, torque is given by \(\tau=r F\). So, if we multiply both sides of the equation above by \(r\), we get torque on the left-hand side. That is,
\[r F=m r^{2} \alpha \nonumber \]
or
\[\tau=m r^{2} \alpha. \nonumber \]
This last equation is the rotational analog of Newton’s second law (\(F=m a\)), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and \(m r^{2}\) is analogous to mass (or inertia). The quantity \(m r^{2}\) is called the rotational inertia or moment of inertia of a point mass \(m\) a distance \(r\) from the center of rotation.
MAKING CONNECTIONS: ROTATIONAL MOTION DYNAMICS
Dynamics for rotational motion is completely analogous to linear or translational dynamics. Dynamics is concerned with force and mass and their effects on motion. For rotational motion, we will find direct analogs to force and mass that behave just as we would expect from our earlier experiences.
Rotational Inertia and Moment of Inertia
Before we can consider the rotation of anything other than a point mass like the one in Figure \(\PageIndex{2}\), we must extend the idea of rotational inertia to all types of objects. To expand our concept of rotational inertia, we define the moment of inertia \(I\) of an object to be the sum of \(m r^{2}\) for all the point masses of which it is composed. That is, \(I=\sum m r^{2}\). Here \(I\) is analogous to \(m\) in translational motion. Because of the distance \(r\), the moment of inertia for any object depends on the chosen axis. Actually, calculating \(I\) is beyond the scope of this text except for one simple case—that of a hoop, which has all its mass at the same distance from its axis. A hoop’s moment of inertia around its axis is therefore \(M R^{2}\), where \(M\) is its total mass and \(R\) its radius. (We use \(M\) and \(R\) for an entire object to distinguish them from \(m\) and \(r\) for point masses.) Figure \(\PageIndex{3}\) shows a variety of formulas of rotational inertia for many different shapes (deriving these formulas require calculus). For our purposes, it is good enough for you to see that rotational inertia is proportional to mass, and that it's proportional to size of object squared. Note that \(I\) has units of mass multiplied by distance squared (\(\mathrm{kg} \cdot \mathrm{m}^{2}\)), as we might expect from its definition.
The general relationship among torque, moment of inertia, and angular acceleration is
\[\text { net } \tau=I \alpha \nonumber \]
or
\[\alpha=\frac{\text { net } \tau}{I}, \nonumber \]
where net \(\tau\) is the total torque from all forces relative to a chosen axis. For simplicity, we will only consider torques exerted by forces in the plane of the rotation. Such torques are either positive or negative and add like ordinary numbers. The relationship in \(\tau=I \alpha\), \(\alpha=\frac{\text { net } \tau}{I}\) is the rotational analog to Newton’s second law and is very generally applicable. This equation is actually valid for any torque, applied to any object, relative to any axis.
As we might expect, the larger the torque is, the larger the angular acceleration is. For example, the harder a child pushes on a merry-go-round, the faster it accelerates. Furthermore, the more massive a merry-go-round, the slower it accelerates for the same torque. The basic relationship between moment of inertia and angular acceleration is that the larger the moment of inertia, the smaller is the angular acceleration. But there is an additional twist. The moment of inertia depends not only on the mass of an object, but also on its distribution of mass relative to the axis around which it rotates. For example, it will be much easier to accelerate a merry-go-round full of children if they stand close to its axis than if they all stand at the outer edge. The mass is the same in both cases, but the moment of inertia is much larger when the children are at the edge.
TAKE-HOME EXPERIMENT
Cut out a circle that has about a 10 cm radius from stiff cardboard. Near the edge of the circle, write numbers 1 to 12 like hours on a clock face. Position the circle so that it can rotate freely about a horizontal axis through its center, like a wheel. (You could loosely nail the circle to a wall.) Hold the circle stationary and with the number 12 positioned at the top, attach a lump of blue putty (sticky material used for fixing posters to walls) at the number 3. How large does the lump need to be to just rotate the circle? Describe how you can change the moment of inertia of the circle. How does this change affect the amount of blue putty needed at the number 3 to just rotate the circle? Change the circle’s moment of inertia and then try rotating the circle by using different amounts of blue putty. Repeat this process several times.
MAKING CONNECTIONS
In statics, the net torque is zero, and there is no angular acceleration. In rotational motion, net torque is the cause of angular acceleration, exactly as in Newton’s second law of motion for rotation.
Exercise \(\PageIndex{1}\)
Torque is the analog of force and moment of inertia is the analog of mass. Force and mass are physical quantities that depend on only one factor. For example, mass is related solely to the numbers of atoms of various types in an object. Are torque and moment of inertia similarly simple?
- Answer
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No. Torque depends on three factors: force magnitude, force direction, and point of application. Moment of inertia depends on both mass and its distribution relative to the axis of rotation. So, while the analogies are precise, these rotational quantities depend on more factors.
Section Summary
- The farther the force is applied from the pivot, the greater is the angular acceleration; angular acceleration is inversely proportional to mass.
- The torque applied on an object is given by lever arm times the applied force: \(\tau=r F\).
- Using the torque, we can discover the rotational version of Newton's Second Law. It says \(\tau=I \alpha\), where \(I\) is rotational inertia, defined for a point mass moving in a circle of radius \(r\) as \(I=m r^{2}\).
- The formula for rotational inertia of an extended body can be complicated. But they are all proportional to the mass of the object and proportional to the square of the size of the object.
Glossary
- torque
- the turning effectiveness of a force
- lever arm
- the perpendicular distance between the center of rotation and the point at which force is being applied
- rotational inertia
- resistance to change of angular velocity; also called moment of inertia ; for a point mass, \(I=m r^{2}\)


2.2: Quadrants
Like a Star Trek quadrant?
First off: Nerd alert!
Secondly, yes, a quadrant is a circle cut into four parts.
What does this have to do with electricity?
Voltage and currents are constantly changing magnitude and direction. When changing direction, they actually rotate in a counterclockwise direction. They are tethered to a point of origin .
Each quadrant contains certain directions.
- Quadrant 1 has 0 to 90 degrees.
- Quadrant 2 has 90 to 180 degrees.
- Quadrant 3 has 180 to 270 degrees.
- Quadrant 4 has 270 to 360 degrees.
This is very important as it helps us to determine which vectors belong in which quadrant.
Polarity
It is also important to understand polarity when dealing with quadrants. A quadrant system is basically an X-Y graph. We use the point of origin as a reference point. On the X axis, anything to the right of the point of origin is positive and anything to the left is negative. On the Y axis, anything above the point of origin is positive and anything underneath it is negative. This means each quadrant has its own polarity, as shown in Figure 30.
This too is extremely important when it comes to adding vectors.
Are you getting excited yet? This is all going to come together in one magical dance.


6.13: Extrasolar Planets
The extrasolar planets tell us a lot about our own solar system’s formation and development. Yet, there are some significant issues at this point. First, there appears to be numerous exoplanets much larger than Jupiter. Why is this so? And many of these extrasolar planets orbit much closer to their stars than our Gas Giants, which are much further from the Sun. The thinking here is that these Gas Giants “migrated” closer to their stars during the development of their stellar systems.
Brown Dwarfs are sub-stellar objects with a mass below that necessary to maintain hydrogen nuclear fusion reactions in their cores, as do stars. Sizes of these brown dwarfs – up to 80 Jupiter masses – range from very large gas giant planets to just below the mass necessary to “turn on” as a star. Are brown dwarfs’ planets? Stars? Failed stars? Two extrasolar planets have been discovered orbiting brown dwarfs.
Consider this…
Giordano Bruno (1548-1600) was an Italian Dominican Friar, poet, philosopher, astronomer, and mathematician. His theories about the Universe were considered extreme for his day, eventually leading to his being burned at the stake. Bruno thought the stars were distant suns being orbited by their own planets. He also thought these planets might even harbor life of its own. Bruno also believed the Universe was infinite.
There are countless suns and countless earths all rotating around their suns…
— Friar Giordano Bruno – (1584)


7.5: Entropy Changes in Reversible Processes
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Learning Objectives
You are expected to be able to define and explain the significance of terms identified in bold .
- A reversible process is one carried out in infinitesimal steps after which, when undone, both the system and surroundings (that is, the world) remain unchanged (see the example of gas expansion-compression below). Although true reversible change cannot be realized in practice, it can always be approximated.
- As a process is carried out in a more reversible manner, the value of w approaches its maximum possible value, and q approaches its minimum possible value.
- Although q is not a state function, the quotient q rev /T is, and is known as the entropy .
A change is said to occur reversibly when it can be carried out in a series of infinitesimal steps, each one of which can be undone by making a similarly minute change to the conditions that bring the change about. For example, the reversible expansion of a gas can be achieved by reducing the external pressure in a series of infinitesimal steps; reversing any step will restore the system and the surroundings to their previous state. Similarly, heat can be transferred reversibly between two bodies by changing the temperature difference between them in infinitesimal steps each of which can be undone by reversing the temperature difference.
The most widely cited example of an irreversible change is the free expansion of a gas into a vacuum. Although the system can always be restored to its original state by recompressing the gas, this would require that the surroundings perform work on the gas. Since the gas does no work on the surrounding in a free expansion (the external pressure is zero, so \(PΔV = 0\),) there will be a permanent change in the surroundings. Another example of irreversible change is the conversion of mechanical work into frictional heat; there is no way, by reversing the motion of a weight along a surface, that the heat released due to friction can be restored to the system.
These diagrams show the same expansion and compression ±ΔV carried out in different numbers of steps ranging from a single step at the top to an "infinite" number of steps at the bottom. As the number of steps increases, the processes become less irreversible; that is, the difference between the work done in expansion and that required to re-compress the gas diminishes. In the limit of an ”infinite” number of steps (bottom), these work terms are identical, and both the system and surroundings (the “world”) are unchanged by the expansion-compression cycle. In all other cases the system (the gas) is restored to its initial state, but the surroundings are forever changed.
Definition: Reversible Changes
A reversible change is one carried out in such as way that, when undone, both the system and surroundings (that is, the world) remain unchanged.
It should go without saying, of course, that any process that proceeds in infinitesimal steps would take infinitely long to occur, so thermodynamic reversibility is an idealization that is never achieved in real processes, except when the system is already at equilibrium, in which case no change will occur anyway! So why is the concept of a reversible process so important?
The answer can be seen by recalling that the change in the internal energy that characterizes any process can be distributed in an infinity of ways between heat flow across the boundaries of the system and work done on or by the system, as expressed by the First Law of thermodynamics
\[ΔU = q + w. \label{first} \]
Each combination of q and w represents a different pathway between the initial and final states. It can be shown that as a process such as the expansion of a gas is carried out in successively longer series of smaller steps, the absolute value of q approaches a minimum, and that of w approaches a maximum that is characteristic of the particular process. Thus when a process is carried out reversibly, the w -term in Equation \ref{first} has its greatest possible value, and the q -term is at its smallest. These special quantities \(w_{max}\) and \(q_{min}\) (which we denote as \(q_{rev}\) and pronounce “q-reversible”) have unique values for any given process and are therefore state functions.
Work and Reversibility
Changes in entropy (\(ΔS\)), together with changes in enthalpy (\(ΔH\)), enable us to predict in which direction a chemical or physical change will occur spontaneously. Before discussing how to do so, however, we must understand the difference between a reversible process and an irreversible one. In a reversible process, every intermediate state between the extremes is an equilibrium state, regardless of the direction of the change. In contrast, an irreversible process is one in which the intermediate states are not equilibrium states, so change occurs spontaneously in only one direction. As a result, a reversible process can change direction at any time, whereas an irreversible process cannot. When a gas expands reversibly against an external pressure such as a piston, for example, the expansion can be reversed at any time by reversing the motion of the piston; once the gas is compressed, it can be allowed to expand again, and the process can continue indefinitely. In contrast, the expansion of a gas into a vacuum (\(P_{ext} = 0\)) is irreversible because the external pressure is measurably less than the internal pressure of the gas. No equilibrium states exist, and the gas expands irreversibly. When gas escapes from a microscopic hole in a balloon into a vacuum, for example, the process is irreversible; the direction of airflow cannot change.
Because work done during the expansion of a gas depends on the opposing external pressure (\(w = P_{ext}ΔV\)), work done in a reversible process is always equal to or greater than work done in a corresponding irreversible process: \(w_{rev} ≥ w_{irrev}\). Whether a process is reversible or irreversible, the first law of thermodynamics holds (Equation \ref{first}). Because \(U\) is a state function, the magnitude of \(ΔU\) does not depend on reversibility and is independent of the path taken. So
\[\begin{align} ΔU &= q_{rev} + w_{rev} \\[4pt] &= q_{irrev} + w_{irrev} \label{Eq1} \end{align}\]
Work done in a reversible process is always equal to or greater than work done in a corresponding irreversible process:
\[w_{rev} \ge w_{irrev}\]
In other words, \(ΔU\) for a process is the same whether that process is carried out in a reversible manner or an irreversible one. We now return to our earlier definition of entropy, using the magnitude of the heat flow for a reversible process (\(q_{rev}\)) to define entropy quantitatively.
For a process that reversibly exchanges a quantity of heat \(q_{rev}\) with the surroundings, the entropy change is defined as
\[ \color{red} \Delta S = \dfrac{q_{rev}}{T} \label{23.2.1}\]
This is the basic way of evaluating \(ΔS\) for constant-temperature processes such as phase changes, or the isothermal expansion of a gas. For processes in which the temperature is not constant such as heating or cooling of a substance, the equation must be integrated over the required temperature range, as discussed below.
Q&A: If no real process can take place reversibly, what use is an expression involving \(q_{rev}\)?
This is a rather fine point that you should understand: although transfer of heat between the system and surroundings is impossible to achieve in a truly reversible manner, this idealized pathway is only crucial for the definition of \(ΔS\); by virtue of its being a state function, the same value of \(ΔS\) will apply when the system undergoes the same net change via any pathway. For example, the entropy change a gas undergoes when its volume is doubled at constant temperature will be the same regardless of whether the expansion is carried out in 1000 tiny steps (as reversible as patience is likely to allow) or by a single-step (as irreversible a pathway as you can get!) expansion into a vacuum (Figure \(\PageIndex{1}\)).
Entropy is an extensive quantity; that is, it is proportional to the quantity of matter in a system; thus 100 g of metallic copper has twice the entropy of 50 g at the same temperature. This makes sense because the larger piece of copper contains twice as many quantized energy levels able to contain the thermal energy.
Entropy and "disorder"
Entropy is still described, particularly in older textbooks, as a measure of disorder. In a narrow technical sense this is correct, since the spreading and sharing of thermal energy does have the effect of randomizing the disposition of thermal energy within a system. But to simply equate entropy with “disorder” without further qualification is extremely misleading because it is far too easy to forget that entropy (and thermodynamics in general) applies only to molecular-level systems capable of exchanging thermal energy with the surroundings. Carrying these concepts over to macro systems may yield compelling analogies, but it is no longer science. It is far better to avoid the term “disorder” altogether in discussing entropy.
\(\Delta S_{sys}\) for an Isothermal Expansion (or Compression)
As a substance becomes more dispersed in space, the thermal energy it carries is also spread over a larger volume, leading to an increase in its entropy. Because entropy, like energy, is an extensive property, a dilute solution of a given substance may well possess a smaller entropy than the same volume of a more concentrated solution, but the entropy per mole of solute (the molar entropy) will of course always increase as the solution becomes more dilute.
For gaseous substances, the volume and pressure are respectively direct and inverse measures of concentration. For an ideal gas that expands at a constant temperature (meaning that it absorbs heat from the surroundings to compensate for the work it does during the expansion), the increase in entropy is given by
\[ \Delta S = R \ln \left( \dfrac{V_2}{V_1} \right) \label{23.2.4}\]
Note: If the gas is allowed to cool during the expansion, the relation becomes more complicated and will best be discussed in a more advanced course.
Because the pressure of an ideal gas is inversely proportional to its volume, i.e.,
\[P = \dfrac{nRT}{V}\]
we can easily alter Equation \ref{23.2.4} to express the entropy change associated with a change in the pressure of an ideal gas:
\[ \Delta S = R \ln \left( \dfrac{P_1}{P_2} \right) \label{23.2.5}\]
Also the concentration \(c=n/V\) for an ideal gas is proportional to pressure
\[P = c RT\]
we can expressing the entropy change directly in concentrations, we have the similar relation
\[ \Delta S = R \ln \left( \dfrac{c_1}{c_2} \right) \label{23.2.6}\]
Although these equations strictly apply only to perfect gases and cannot be used at all for liquids and solids, it turns out that in a dilute solution, the solute can often be treated as a gas dispersed in the volume of the solution, so the last equation can actually give a fairly accurate value for the entropy of dilution of a solution. We will see later that this has important consequences in determining the equilibrium concentrations in a homogeneous reaction mixture.
The Relationship between Internal Energy and Entropy
Because the quantity of heat transferred (\(q_{rev}\)) is directly proportional to the absolute temperature of an object (\(T\)) (\(q_{rev} \propto T\)), the hotter the object, the greater the amount of heat transferred. Moreover, adding heat to a system increases the kinetic energy of the component atoms and molecules and hence their disorder (\(ΔS ∝ q_{rev}\)). Combining these relationships for any reversible process,
\[q_{\textrm{rev}}=T\Delta S\]
and
Because the numerator (q rev ) is expressed in units of energy (joules), the units of ΔS are joules/kelvin (J/K). Recognizing that the work done in a reversible process at constant pressure is
\[w_{rev} = −PΔV,\]
we can express Equation \(\ref{Eq1}\) as follows:
\[\begin{align} ΔU &= q_{rev} + w_{rev} \\[4pt] &= TΔS − PΔV \label{Eq3} \end{align} \]
Thus the change in the internal energy of the system is related to the change in entropy, the absolute temperature, and the \(PV\) work done.
To illustrate the use of Equation \(\ref{Eq2}\) and Equation \(\ref{Eq3}\), we consider two reversible processes before turning to an irreversible process. When a sample of an ideal gas is allowed to expand reversibly at constant temperature, heat must be added to the gas during expansion to keep its T constant (Figure \(\PageIndex{5}\)). The internal energy of the gas does not change because the temperature of the gas does not change; that is, ΔU = 0 and q rev = −w rev . During expansion, ΔV > 0, so the gas performs work on its surroundings:
\[w_{rev} = −PΔV < 0.\]
According to Equation \(\ref{Eq3}\), this means that \(q_{rev}\) must increase during expansion; that is, the gas must absorb heat from the surroundings during expansion, and the surroundings must give up that same amount of heat. The entropy change of the system is therefore
\[ΔS_{sys} = +\dfrac{q_{rev}}{T}\]
and the entropy change of the surroundings is
\[ΔS_{surr} = −\dfrac{q_{rev}}{T}.\]
The corresponding change in entropy of the universe is then as follows:
\[\begin{align} \Delta S_{\textrm{univ}} &=\Delta S_{\textrm{sys}}+\Delta S_{\textrm{surr}} \\[4pt] &=\dfrac{q_{\textrm{rev}}}{T}+\left(-\dfrac{q_\textrm{rev}}{T}\right) \\[4pt] &= 0 \label{Eq4} \end{align} \]
Thus no change in \(ΔS_{univ} \) has occurred.
In the initial state (top), the temperatures of a gas and the surroundings are the same. During the reversible expansion of the gas, heat must be added to the gas to maintain a constant temperature. Thus the internal energy of the gas does not change, but work is performed on the surroundings. In the final state (bottom), the temperature of the surroundings is lower because the gas has absorbed heat from the surroundings during expansion.
Now consider the reversible melting of a sample of ice at 0°C and 1 atm. The enthalpy of fusion of ice is 6.01 kJ/mol, which means that 6.01 kJ of heat are absorbed reversibly from the surroundings when 1 mol of ice melts at 0°C, as illustrated in Figure \(\PageIndex{6}\). The surroundings constitute a sample of low-density carbon foam that is thermally conductive, and the system is the ice cube that has been placed on it. The direction of heat flow along the resulting temperature gradient is indicated with an arrow. From Equation \(\ref{Eq2}\), we see that the entropy of fusion of ice can be written as follows:
\[\Delta S_{\textrm{fus}}=\dfrac{q_{\textrm{rev}}}{T}=\dfrac{\Delta H_{\textrm{fus}}}{T} \label{Eq5}\]
By convention, a thermogram shows cold regions in blue, warm regions in red, and thermally intermediate regions in green. When an ice cube (the system, dark blue) is placed on the corner of a square sample of low-density carbon foam with very high thermal conductivity, the temperature of the foam is lowered (going from red to green). As the ice melts, a temperature gradient appears, ranging from warm to very cold. An arrow indicates the direction of heat flow from the surroundings (red and green) to the ice cube. The amount of heat lost by the surroundings is the same as the amount gained by the ice, so the entropy of the universe does not change.
In this case,
\[\begin{align} ΔS_{fus} &= \dfrac{6.01\, kJ/mol}{273\, K} \\[4pt] &= 22.0\, J/(mol•K) \\[4pt] &= ΔS_{sys}.\end{align}\]
The amount of heat lost by the surroundings is the same as the amount gained by the ice, so
\[\begin{align} ΔS_{surr} &= \dfrac{q_{rev}}{T} \\[4pt] &= \dfrac{ −6.01\, kJ/mol}{273\, K} \\[4pt] &= −22.0\, J/(mol•K).\end{align}\]
Once again, we see that the entropy of the universe does not change:
\[\begin{align} ΔS_{univ} &= ΔS_{sys} + ΔS_{surr} \\[4pt] &= 22.0 \,J/(mol•K) − 22.0\, J/(mol•K) \\[4pt] &= 0 \end{align}\]
In these two examples of reversible processes, the entropy of the universe is unchanged. This is true of all reversible processes and constitutes part of the second law of thermodynamics: the entropy of the universe remains constant in a reversible process, whereas the entropy of the universe increases in an irreversible (spontaneous) process.
The Second Law of Thermodynamics
The entropy of the universe increases during a spontaneous process. It also increases during an observable non-spontaneous process.
As an example of an irreversible process, consider the entropy changes that accompany the spontaneous and irreversible transfer of heat from a hot object to a cold one, as occurs when lava spewed from a volcano flows into cold ocean water. The cold substance, the water, gains heat (q > 0), so the change in the entropy of the water can be written as ΔS cold = q/T cold . Similarly, the hot substance, the lava, loses heat (q < 0), so its entropy change can be written as ΔS hot = −q/T hot , where T cold and T hot are the temperatures of the cold and hot substances, respectively. The total entropy change of the universe accompanying this process is therefore
\[\begin{align} \Delta S_{\textrm{univ}} &= \Delta S_{\textrm{cold}}+\Delta S_{\textrm{hot}} \\[4pt] &= \dfrac{q}{T_{\textrm{cold}}}+\left(-\dfrac{q}{T_{\textrm{hot}}}\right) \label{Eq6}\end{align}\]
The numerators on the right side of Equation \(\ref{Eq6}\) are the same in magnitude but opposite in sign. Whether ΔS univ is positive or negative depends on the relative magnitudes of the denominators. By definition, T hot > T cold , so −q/T hot must be less than q/T cold , and ΔS univ must be positive. As predicted by the second law of thermodynamics, the entropy of the universe increases during this irreversible process. Any process for which ΔS univ is positive is, by definition, a spontaneous one that will occur as written. Conversely, any process for which ΔS univ approaches zero will not occur spontaneously as written but will occur spontaneously in the reverse direction. We see, therefore, that heat is spontaneously transferred from a hot substance, the lava, to a cold substance, the ocean water. In fact, if the lava is hot enough (e.g., if it is molten), so much heat can be transferred that the water is converted to steam (Figure \(\PageIndex{7}\)).
When molten lava flows into cold ocean water, so much heat is spontaneously transferred to the water that steam is produced.
Example \(\PageIndex{1}\): Tin Pest
Tin has two allotropes with different structures. Gray tin (α-tin) has a structure similar to that of diamond, whereas white tin (β-tin) is denser, with a unit cell structure that is based on a rectangular prism. At temperatures greater than 13.2°C, white tin is the more stable phase, but below that temperature, it slowly converts reversibly to the less dense, powdery gray phase. This phenomenon was argued to plagued Napoleon’s army during his ill-fated invasion of Russia in 1812: the buttons on his soldiers’ uniforms were made of tin and disintegrated during the Russian winter, adversely affecting the soldiers’ health (and morale). The conversion of white tin to gray tin is exothermic, with ΔH = −2.1 kJ/mol at 13.2°C.
- What is ΔS for this process?
- Which is the more highly ordered form of tin—white or gray?
Given : ΔH and temperature
Asked for: ΔS and relative degree of order
Strategy :
Use Equation \(\ref{Eq2}\) to calculate the change in entropy for the reversible phase transition. From the calculated value of ΔS, predict which allotrope has the more highly ordered structure.
Solution
- We know from Equation \(\ref{Eq2}\) that the entropy change for any reversible process is the heat transferred (in joules) divided by the temperature at which the process occurs. Because the conversion occurs at constant pressure, and ΔH and ΔU are essentially equal for reactions that involve only solids, we can calculate the change in entropy for the reversible phase transition where q rev = ΔH. Substituting the given values for ΔH and temperature in kelvins (in this case, T = 13.2°C = 286.4 K),
- The fact that ΔS < 0 means that entropy decreases when white tin is converted to gray tin. Thus gray tin must be the more highly ordered structure.
Note: Whether failing buttons were indeed a contributing factor in the failure of the invasion remains disputed; critics of the theory point out that the tin used would have been quite impure and thus more tolerant of low temperatures. Laboratory tests provide evidence that the time required for unalloyed tin to develop significant tin pest damage at lowered temperatures is about 18 months, which is more than twice the length of Napoleon's Russian campaign. It is clear though that some of the regiments employed in the campaign had tin buttons and that the temperature reached sufficiently low values (at least -40 °C) to facilitate tin pest.
Exercise \(\PageIndex{1}\): Sulfur
Elemental sulfur exists in two forms: an orthorhombic form (S α ), which is stable below 95.3°C, and a monoclinic form (S β ), which is stable above 95.3°C. The conversion of orthorhombic sulfur to monoclinic sulfur is endothermic, with \(ΔH = 0.401\, kJ/mol\) at 1 atm.
- What is ΔS for this process?
- Which is the more highly ordered form of sulfur—S α or S β ?
- Answer a
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1.09 J/(mol•K)
- Answer b
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S α
Summary
During a spontaneous process, the entropy of the universe increases. \[\Delta S=\dfrac{q_{\textrm{rev}}}{T}\nonumber\] A reversible process is one for which all intermediate states between extremes are equilibrium states ; it can change direction at any time. In contrast, an irreversible process occurs in one direction only. The change in entropy of the system or the surroundings is the quantity of heat transferred divided by the temperature. The second law of thermodynamics states that in a reversible process, the entropy of the universe is constant, whereas in an irreversible process, such as the transfer of heat from a hot object to a cold object, the entropy of the universe increases.


10.3: Find the Equation of a Line
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Learning Objectives
By the end of this section, you will be able to:
- Find an equation of the line given the slope and y-intercept
- Find an equation of the line given the slope and a point
- Find an equation of the line given two points
- Find an equation of a line parallel to a given line
- Find an equation of a line perpendicular to a given line
Note
Before you get started, take this readiness quiz.
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Solve: \(\frac{2}{3} = \frac{x}{5}\).
If you missed this problem, review Exercise 2.2.4 . -
Simplify: \(−\frac{2}{5}(x−15)\).
If you missed this problem, review Exercise 1.10.34 .
How do online retailers know that ‘you may also like’ a particular item based on something you just ordered? How can economists know how a rise in the minimum wage will affect the unemployment rate? How do medical researchers create drugs to target cancer cells? How can traffic engineers predict the effect on your commuting time of an increase or decrease in gas prices? It’s all mathematics.
You are at an exciting point in your mathematical journey as the mathematics you are studying has interesting applications in the real world.
The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. Data is collected and graphed. If the data points appear to form a straight line, an equation of that line can be used to predict the value of one variable based on the value of the other variable.
To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this section we will look at several ways to write the equation of a line. The specific method we use will be determined by what information we are given.
Find an Equation of the Line Given the Slope and y -Intercept
We can easily determine the slope and intercept of a line if the equation was written in slope–intercept form, y=mx+b. Now, we will do the reverse—we will start with the slope and y -intercept and use them to find the equation of the line.
Example \(\PageIndex{1}\)
Find an equation of a line with slope −7 and y -intercept (0,−1).
Solution
Since we are given the slope and y -intercept of the line, we can substitute the needed values into the slope–intercept form, y=mx+b.
| Name the slope. | |
| Name the y -intercept. | |
| Substitute the values into y=mx+b. | |
Try It \(\PageIndex{2}\)
Find an equation of a line with slope \(\frac{2}{5}\) and y -intercept (0,4).
- Answer
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\(y = \frac{2}{5}x + 4\)
Try It \(\PageIndex{3}\)
Find an equation of a line with slope −1 and y -intercept (0,−3).
- Answer
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\(y=−x−3\)
Sometimes, the slope and intercept need to be determined from the graph.
Example \(\PageIndex{4}\)
Find the equation of the line shown.
Solution
We need to find the slope and y -intercept of the line from the graph so we can substitute the needed values into the slope–intercept form, \(y=mx+b\).
To find the slope, we choose two points on the graph.
The y -intercept is (0,−4) and the graph passes through (3,−2).
| Find the slope by counting the rise and run. | |
| Find the y -intercept. | |
| Substitute the values into y=mx+b. | |
Try It \(\PageIndex{5}\)
Find the equation of the line shown in the graph.
- Answer
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\(y=\frac{3}{5}x+1\)
Try It \(\PageIndex{6}\)
Find the equation of the line shown in the graph.
- Answer
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\(y=\frac{4}{3}x−5\)
Find an Equation of the Line Given the Slope and a Point
Finding an equation of a line using the slope–intercept form of the equation works well when you are given the slope and y -intercept or when you read them off a graph. But what happens when you have another point instead of the y -intercept?
We are going to use the slope formula to derive another form of an equation of the line. Suppose we have a line that has slope mm and that contains some specific point \((x_{1}, y_{1})\) and some other point, which we will just call (x,y). We can write the slope of this line and then change it to a different form.
\(\begin{array} {lrll}&m &=\frac{y-y_{1}}{x-x_{1}} \\ \text{Multiply both sides of the equation by }x−x_{1}.&m\left(x-x_{1}\right) &=\left(\frac{y-y_{1}}{x-x_{1}}\right)\left(x-x_{1}\right) \\ \text{Simplify.}&m\left(x-x_{1}\right) &=y-y_{1} \\ \text{Rewrite the equation with the y terms on the left.} &y-y_{1} &=m\left(x-x_{1}\right) \end{array}\)
This format is called the point–slope form of an equation of a line.
POINT–SLOPE FORM OF AN EQUATION OF A LINE
The point–slope form of an equation of a line with slope \(m\) and containing the point \((x_{1}, y_{1})\) is
We can use the point–slope form of an equation to find an equation of a line when we are given the slope and one point. Then we will rewrite the equation in slope–intercept form. Most applications of linear equations use the the slope–intercept form.
Example \(\PageIndex{7}\): Find an Equation of a Line Given the Slope and a Point
Find an equation of a line with slope \(m=\frac{2}{5}\) that contains the point (10,3). Write the equation in slope–intercept form.
Solution
Try It \(\PageIndex{8}\)
Find an equation of a line with slope \(m=\frac{5}{6}\) and containing the point (6,3).
- Answer
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\(y=\frac{5}{6}x−2\)
Try It \(\PageIndex{9}\)
Find an equation of a line with slope \(m=\frac{2}{3}\) and containing the point (9,2).
- Answer
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\(y=\frac{2}{3}x−4\)
FIND AN EQUATION OF A LINE GIVEN THE SLOPE AND A POINT.
- Identify the slope.
- Identify the point.
- Substitute the values into the point-slope form, \(y−y_{1}=m(x−x_{1})\).
- Write the equation in slope–intercept form.
Example \(\PageIndex{10}\)
Find an equation of a line with slope \(m=−\frac{1}{3}\) that contains the point (6,−4). Write the equation in slope–intercept form.
Solution
Since we are given a point and the slope of the line, we can substitute the needed values into the point–slope form, \(y−y_{1}=m(x−x_{1})\).
| Identify the slope. | |
| Identify the point. | |
| Substitute the values into \(y−y_{1}=m(x−x_{1})\). | |
| Simplify. | |
| Write in slope–intercept form. |
Try It \(\PageIndex{11}\)
Find an equation of a line with slope \(m=−\frac{2}{5}\) and containing the point (10,−5).
- Answer
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\(y=−\frac{2}{5}x−1\)
Try It \(\PageIndex{12}\)
Find an equation of a line with slope \(m=−\frac{3}{4}\), and containing the point (4,−7).
- Answer
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\(y=−\frac{3}{4}x−4\)
Example \(\PageIndex{13}\)
Find an equation of a horizontal line that contains the point (−1,2). Write the equation in slope–intercept form.
Solution
Every horizontal line has slope 0. We can substitute the slope and points into the point–slope form, \(y−y_{1}=m(x−x_{1})\).
| Identify the slope. | |
| Identify the point. | |
| Substitute the values into \(y−y_{1}=m(x−x_{1})\). | |
| Simplify. | |
| Write in slope–intercept form. | It is in y -form, but could be written y=0x+2. |
Try It \(\PageIndex{14}\)
Find an equation of a horizontal line containing the point (−3,8).
- Answer
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y = 8
Try It \(\PageIndex{15}\)
Find an equation of a horizontal line containing the point (−1,4).
- Answer
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y = 4
Find an Equation of the Line Given Two Points
When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.
We have two options so far for finding an equation of a line: slope–intercept or point–slope. Since we will know two points, it will make more sense to use the point–slope form.
But then we need the slope. Can we find the slope with just two points? Yes. Then, once we have the slope, we can use it and one of the given points to find the equation.
Example \(\PageIndex{16}\): Find an Equation of a Line Given Two Points
Find an equation of a line that contains the points (5,4) and (3,6). Write the equation in slope–intercept form.
Solution
Use the point (3,6) and see that you get the same equation.
Try It \(\PageIndex{17}\)
Find an equation of a line containing the points (3,1) and (5,6).
- Answer
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\(y=\frac{5}{2}x−\frac{13}{2}\)
Try It \(\PageIndex{18}\)
Find an equation of a line containing the points (1,4) and (6,2).
- Answer
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\(y=−\frac{2}{5}x+\frac{22}{5}\)
FIND AN EQUATION OF A LINE GIVEN TWO POINTS.
- Find the slope using the given points.
- Choose one point.
- Substitute the values into the point-slope form, \(y−y_{1}=m(x−x_{1})\).
- Write the equation in slope–intercept form.
Example \(\PageIndex{19}\)
Find an equation of a line that contains the points (−3,−1) and (2,−2). Write the equation in slope–intercept form.
Solution
Since we have two points, we will find an equation of the line using the point–slope form. The first step will be to find the slope.
| Find the slope of the line through (−3, −1) and (2, −2). | |
| Choose either point. | |
| Substitute the values into \(y−y_{1}=m(x−x_{1})\). | |
| Write in slope–intercept form. |
Try It \(\PageIndex{20}\)
Find an equation of a line containing the points (−2,−4) and (1,−3).
- Answer
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\(y=\frac{1}{3}x−\frac{10}{3}\)
Try It \(\PageIndex{21}\)
Find an equation of a line containing the points (−4,−3) and (1,−5).
- Answer
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\(y=−\frac{2}{5}x−\frac{23}{5}\)
Example \(\PageIndex{22}\)
Find an equation of a line that contains the points (−2,4) and (−2,−3). Write the equation in slope–intercept form.
Solution
Again, the first step will be to find the slope.
\(\begin{array}{lrl} \text { Find the slope of the line through }(-2,4) \text { and }(-2,-3) & & &\\ &m &=&\frac{y_{2}-x_{1}}{x_{2}-x_{1}} \\ &m &=&\frac{-3-4}{-2-(-2)} \\ &m &= &\frac{-7}{0} \\ \\ \text { The slope is undefined. } & & &\end{array}\)
This tells us it is a vertical line. Both of our points have an x -coordinate of −2. So our equation of the line is x=−2. Since there is no yy, we cannot write it in slope–intercept form.
You may want to sketch a graph using the two given points. Does the graph agree with our conclusion that this is a vertical line?
Try It \(\PageIndex{23}\)
Find an equation of a line containing the points (5,1) and (5,−4).
- Answer
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x = 5
Try It \(\PageIndex{24}\)
Find an equation of a line containing the points (−4,4) and (−4,3).
- Answer
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x=−4
We have seen that we can use either the slope–intercept form or the point–slope form to find an equation of a line. Which form we use will depend on the information we are given. This is summarized in Table \(\PageIndex{1}\).
| To Write an Equation of a Line | ||
| If given: | Use: | Form: |
| Slope and y -intercept | slope–intercept | y=mx+b |
| Slope and a point | point–slope | \(y−y_{1}=m(x−x_{1})\) |
| Two points | point–slope | \(y−y_{1}=m(x−x_{1})\) |
Find an Equation of a Line Parallel to a Given Line
Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point–slope equation.
First let’s look at this graphically.
The graph shows the graph of y=2x−3. We want to graph a line parallel to this line and passing through the point (−2,1).
We know that parallel lines have the same slope. So the second line will have the same slope as y=2x−3. That slope is \(m_{\|} = 2\). We’ll use the notation \(m_{\|}\) to represent the slope of a line parallel to a line with slope m. (Notice that the subscript ∥ looks like two parallel lines.)
The second line will pass through (−2,1) and have m=2. To graph the line, we start at (−2,1) and count out the rise and run. With m=2 (or \(m=\frac{2}{1}\)), we count out the rise 2 and the run 1. We draw the line.
Do the lines appear parallel? Does the second line pass through (−2,1)?
Now, let’s see how to do this algebraically.
We can use either the slope–intercept form or the point–slope form to find an equation of a line. Here we know one point and can find the slope. So we will use the point–slope form.
Example \(\PageIndex{25}\): How to Find an Equation of a Line Parallel to a Given Line
Find an equation of a line parallel to y=2x−3 that contains the point (−2,1). Write the equation in slope–intercept form.
Solution
Does this equation make sense? What is the y -intercept of the line? What is the slope?
Try It \(\PageIndex{26}\)
Find an equation of a line parallel to the line y=3x+1 that contains the point (4,2). Write the equation in slope–intercept form.
- Answer
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y=3x−10
Try It \(\PageIndex{27}\)
Find an equation of a line parallel to the line \(y=\frac{1}{2}x−3\) that contains the point (6,4).
- Answer
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\(y=\frac{1}{2}x+1\)
FIND AN EQUATION OF A LINE PARALLEL TO A GIVEN LINE.
- Find the slope of the given line.
- Find the slope of the parallel line.
- Identify the point.
- Substitute the values into the point–slope form, \(y−y_{1}=m(x−x_{1})\).
- Write the equation in slope–intercept form.
Find an Equation of a Line Perpendicular to a Given Line
Now, let’s consider perpendicular lines. Suppose we need to find a line passing through a specific point and which is perpendicular to a given line. We can use the fact that perpendicular lines have slopes that are negative reciprocals. We will again use the point–slope equation, like we did with parallel lines.
The graph shows the graph of y=2x−3. Now, we want to graph a line perpendicular to this line and passing through (−2,1).
We know that perpendicular lines have slopes that are negative reciprocals. We’ll use the notation \(m_{\perp}\) to represent the slope of a line perpendicular to a line with slope m. (Notice that the subscript \(_{\perp}\) looks like the right angles made by two perpendicular lines.)
\[\begin{array}{cl}{y=2 x-3} & {\text { perpendicular line }} \\ {m=2} & {m_{\perp}=-\frac{1}{2}}\end{array}\]
We now know the perpendicular line will pass through (−2,1) with \(m_{\perp}=−\frac{1}{2}\).
To graph the line, we will start at (−2,1) and count out the rise −1 and the run 2. Then we draw the line.
Do the lines appear perpendicular? Does the second line pass through (−2,1)?
Now, let’s see how to do this algebraically. We can use either the slope–intercept form or the point–slope form to find an equation of a line. In this example we know one point, and can find the slope, so we will use the point–slope form.
Example \(\PageIndex{28}\)
Find an equation of a line perpendicular to y=2x−3 that contains the point (−2,1). Write the equation in slope–intercept form.
Solution
Try It \(\PageIndex{29}\)
Find an equation of a line perpendicular to the line y=3x+1 that contains the point (4,2). Write the equation in slope–intercept form.
- Answer
-
\(y=−\frac{1}{3}x+\frac{10}{3}\)
Try It \(\PageIndex{30}\)
Find an equation of a line perpendicular to the line \(y=\frac{1}{2}x−3\) that contains the point (6,4).
- Answer
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y=−2x+16
FIND AN EQUATION OF A LINE PERPENDICULAR TO A GIVEN LINE.
- Find the slope of the given line.
- Find the slope of the perpendicular line.
- Identify the point.
- Substitute the values into the point–slope form, \(y−y_{1}=m(x−x_{1})\).
- Write the equation in slope–intercept form.
Example \(\PageIndex{31}\)
Find an equation of a line perpendicular to x=5 that contains the point (3,−2). Write the equation in slope–intercept form.
Solution
Again, since we know one point, the point–slope option seems more promising than the slope–intercept option. We need the slope to use this form, and we know the new line will be perpendicular to x=5. This line is vertical, so its perpendicular will be horizontal. This tells us the \(m_{\perp}=0\).
\(\begin{array}{lrll}{\text { Identify the point. }} &{(3}&{,}&{-2)}\\ {\text { Identify the slope of the perpendicular line. }} & {m_{\perp}}&{=}&{0} \\ {\text { Substitute the values into } y-y_{1}=m\left(x-x_{1}\right) .} & {y-y_{1}}&{=}&{m\left(x-x_{1}\right)} \\{} &{y−(−2)}&{=}&{0(x−3)} \\{\text { Simplify. }} & {y+2}&{=}&{0} \\ &{y}&{=}&{-2}\end{array}\)
Sketch the graph of both lines. Do they appear to be perpendicular?
Try It \(\PageIndex{32}\)
Find an equation of a line that is perpendicular to the line x=4 that contains the point (4,−5). Write the equation in slope–intercept form.
- Answer
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y=−5
Try It \(\PageIndex{33}\)
Find an equation of a line that is perpendicular to the line x=2 that contains the point (2,−1). Write the equation in slope–intercept form.
- Answer
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y=−1
In Example \(\PageIndex{31}\), we used the point–slope form to find the equation. We could have looked at this in a different way.
We want to find a line that is perpendicular to x=5 that contains the point (3,−2). The graph shows us the line x=5 and the point (3,−2).
We know every line perpendicular to a vertical line is horizontal, so we will sketch the horizontal line through (3,−2).
Do the lines appear perpendicular?
If we look at a few points on this horizontal line, we notice they all have y -coordinates of −2. So, the equation of the line perpendicular to the vertical line x=5 is y=−2.
Example \(\PageIndex{34}\)
Find an equation of a line that is perpendicular to y=−4 that contains the point (−4,2). Write the equation in slope–intercept form.
Solution
The line y=−4 is a horizontal line. Any line perpendicular to it must be vertical, in the form x=a. Since the perpendicular line is vertical and passes through (−4,2), every point on it has an x -coordinate of −4. The equation of the perpendicular line is x=−4. You may want to sketch the lines. Do they appear perpendicular?
Try It \(\PageIndex{35}\)
Find an equation of a line that is perpendicular to the line y=1 that contains the point (−5,1). Write the equation in slope–intercept form.
- Answer
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x=−5
Try It \(\PageIndex{36}\)
Find an equation of a line that is perpendicular to the line y=−5 that contains the point (−4,−5).
- Answer
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x=−4
Note
Access this online resource for additional instruction and practice with finding the equation of a line.
Key Concepts
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To Find an Equation of a Line Given the Slope and a Point
- Identify the slope.
- Identify the point.
- Substitute the values into the point-slope form, \(y−y_{1}=m(x−x_{1})\).
- Write the equation in slope-intercept form.
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To Find an Equation of a Line Given Two Points
- Find the slope using the given points.
- Choose one point.
- Substitute the values into the point-slope form, \(y−y_{1}=m(x−x_{1})\).
- Write the equation in slope-intercept form.
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To Write an Equation of a Line
- If given slope and \(y\)-intercept, use slope–intercept form \(y=mx+b\).
- If given slope and a point, use point–slope form \(y−y_{1}=m(x−x_{1})\).
- If given two points, use point–slope form \(y−y_{1}=m(x−x_{1})\).
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To Find an Equation of a Line Parallel to a Given Line
- Find the slope of the given line.
- Find the slope of the parallel line.
- Identify the point.
- Substitute the values into the point-slope form, \(y−y_{1}=m(x−x_{1})\).
- Write the equation in slope-intercept form.
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To Find an Equation of a Line Perpendicular to a Given Line
- Find the slope of the given line.
- Find the slope of the perpendicular line.
- Identify the point.
- Substitute the values into the point-slope form, \(y−y_{1}=m(x−x_{1})\).
- Write the equation in slope-intercept form.
Glossary
- point–slope form
- The point–slope form of an equation of a line with slope mm and containing the point \(\left(x_{1}, y_{1}\right)\) is \(y-y_{1}=m\left(x-x_{1}\right)\).