1.3E: Describing Relationships With Linear Functions (Exercises)
Section 1.3 Exercises
1. Which of the following tables could represent a linear function?
2. Which of the following tables could represent a linear function?
3. Suppose a patient's heart rate was measured after different doses of a trial medication was administered as detailed in the table below. Determine if the heart rate increasing linearly relative to the dosage. Justify your conclusion.
| Dosage (in mg) | 0 | 5 | 10 | 20 |
| Heart Rate (in bpm) | 80 | 100 | 115 | 130 |
4. Suppose a steel cylindrical steel alloy bar is being pulled in tension along its length with varying loads. The amount of the bar stretches (deflection) is measured. The stress on the steel is the amount of force applied per square unit of cross-sectional area. The strain on the material is the amount of deflection (or stretching) of the material per unit in length. The stress and strain for the steel alloy bar is shown below. Determine if the heart rate increasing linearly relative to the dosage. Justify your conclusion.
| Strain (in inches per inch) | 0 | 5 | 10 | 15 | 20 | 30 |
| Stress (in pounds per square inch) | 0 | 14229 | 28508 | 42737 | 56916 | 85374 |
5. Find the slope of the line that passes through the points (2, 4) and (4, 10).
6. Find the slope of the line that passes through the points (-1, 4) and (5, 2).
7. Find the slope of the lines graphed below.
a) b)
8. The Consumer Price Index (CPI) measure the 12-month percentage change in the price of various items. The 12-month percentage change (or CPI) in Energy in the US was 32.95% in July 2022 and dropped to 17.6% by October 2022 (as reported by the US Bureau of Labor Statistics at https://www.bls.gov/opub on July 29, 2024). The CPI was decreasing at a relatively constant rate per month in this time period. Find the slope of the line connecting these data points and interpret it as a rate of change.
9. The 12-month percentage change (or CPI) for all items in the US was 6.2% in October 2021 and increased to 8.6% by May 2022 (as reported by the US Bureau of Labor Statistics at https://www.bls.gov/opub on July 29, 2024). The CPI was increasing at a relatively constant rate per month in this time period. Find the slope of the line connecting these data points and interpret it as a rate of change.
10. Determine the slope and y-intercept of the line \(f(x) = 4x + 3\). Is the function is increasing or decreasing?
11. Determine the slope and y-intercept of the line \(2x-7y=1\). Is the function is increasing or decreasing?
12. Determine the slope and y-intercept of the line \(3x+5y=8\). Is the function is increasing or decreasing?
13. Find the equation of a line with a slope \(m=3\) and \(y\)-intercept at (0,-2).
14. Find the equation of a line with a slope \(m=2\) that contains the point (3,-4).
15. Find the equation of a line that contains the points (1,2) and (4,7).
16. Find the equation of the linear function \(f\) where \(f(-5) = -4\) and \(f(5) = 2\).
17. Find the equation of the linear function with an \(x\)-intercept at (-2, 0) and \(y\)-intercept at (0, -3).
18. Find an equation for the function graphed below.
a) b)
19. Find an equation for the function graphed below.
a) b)
20. Maria is climbing a mountain. Maria's elevation, \(E(t)\), in feet after \(t\) minutes is given by \(E(t) =1200 + 40t\). Write a complete sentence describing Maria’s starting elevation and how it is changing over time.
21. The population of Reedley, California was 24,774 in 2012. Over the next 10 years the population grew by approximately 67 people each year (as reported by the US Census Beareau at www.census.gov on July 31, 2024).
a) Write an equation, \(P=f(t)\) for the population of Reedley \(t\) years since 2010.
b) Predict the population of Reedley in 2025 if this trend continues.
22. A boat is 100 miles away from the marina, sailing directly towards it at 10 miles per hour.
a) Write an equation for the distance of the boat \(d\) from the marina after \(t\) hours.
b) How long until the boat is 7 miles from the marina?
23. A gym membership with two personal training sessions costs $125, while gym membership with 5 personal training sessions costs $260. Assume the cost increases at a constant rate.
a) Write an equation, \(C=f(s)\) for the cost of \(s\) training sessions.
b) Interpret the meaning of the slope in this application.
c) Predict the cost of 9 sessions.
24. In the school year beginning in the fall of 2021 there were 13,061 students classified as English Language Learners in the Fresno Unified School District. By the fall 2023 school year, there were 14,432 English Language Learners (as reported by the Ed-Data Education Data Partnership at www.ed-data.org on July 31, 2024). Assume the number of English language learners is increasing at a constant rate.
a) Write an equation, \(L=f(t)\) for the number of English language learners in the Fresno Unified School District \(t\) years since 2021.
b) Interpret the meaning of the slope in this application.
c) Using your equation, predict the number of English language learners in 2028 if this trend continues.
25. A phone company has a monthly cellular data plan where a customer pays a flat monthly fee and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly cost will be $17.80.
a) Write a linear equation for the monthly cost \(C =f(x)\) of the data plan as a function of \(x\), the number of MB used.
b) Interpret the meaning of the slope in this application.
c) Predict the total monthly cost if 250 MB are used.
26. The world’s proven oil reserves was 223.27 billion barrels in 2010. The proven oil reserves grew tp 236.81 billion in 2018(as reported by Our World In Data at ourworldindata.org on July 31, 2024). Assume the proven oil reserves increases at a constant rate in this time period.
a. Find an equation for the proven oil reserves, \(R\), in terms of \(t\), the number of years since 2010.
b. Interpret the meaning of the slope in this application.
c. Predict the proven oil reserves in 2028 if this trend continues.
27. The average hourly wage of a worker in Fresno County was $21.08 in 2015. By 2020, the average hourly wage was $25.08 (as reported by the U.S. Bureau of Labor Statistics at https://www.bls.gov/regions/west/news-release on July 24, 2024). Assume the average hourly wage, W, increases at a constant rate relative to time. Let \(W\) represent the average hourly wage of a worker in Fresno County \(t\) years since 2015.
a) Find a linear function \(W=f(t)\) for the average hourly wage of a worker in Fresno County \(t\) years since 2015.
b) Interpret the meaning of the slope in the application.
c) Predict the average hourly wage in 2025 if the average hourly wage continues to increase at a constant rate.
28. Suppose in a study on endurance, a person with a heart rate of 100 beats per minute (bpm) consumed 1 liter of oxygen per minute. When the same person's heart rate was raised to 160 bpm, they consumed 2 liters of oxygen per minute. Assume the oxygen consumption increases at a constant rate relative to heart rate.
a. Find a formula for a linear function \(C=f(h)\) that models the oxygen consumption \(C\) in terms of the heart rate \(h\).
b. Interpret the meaning of the slope in this application.
c. Use your function to predict the oxygen consumption when the heart tate is 198 bpm.
- Answer
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1. The functions \(f\) and \(g\) represent linear functions since their rates of change are constant.
3. No, the patient's heart rate is not increasing linearly relative to dosage since the rate of increase is not constant. On the interval [0,5], \(\dfrac{\Delta h}{\Delta d} = 4\) whereas on the interval [5,10], \(\dfrac{\Delta h}{\Delta d} = 3\).
5. \(m=3\)
7. (a) \(m=\dfrac{2}{3}\) (b) \(m=\dfrac{-5}{4}\)
9. \(m \approx 0.343\). The CPI (the 12-month percentage change in price) of all goods increases by 0.343% per month.
11. \(m=\dfrac{2}{7}\). The function is increasing.
13. \(y=3x-2\)
15. \(y=\dfrac{5}{3}x+\dfrac{1}{3}\)
17. \(y=\dfrac{3}{2}x-3\)
19. (a) \(y=-2x+3\) (b) \(y=3x+1\)
21. (a) \(P=f(t)=67t+24,774\) (b) \(P=f(13)=25,645\)
23. (a) \(C=f(s)=45s+35\) (b) The cost is increasing by $45 per session. (c) Nine sessions will cost $440.
25. (a) \(C=f(x)=0.06x+10\) (b) The cost is increasing by $0.06 per MB of data. (c) 250 MB of data will cost $25.
27. (a) \(W=f(t)=0.8t+21.08\) (b) The average hourly wage is increasing by $0.80 per yer from 2015 to 2020. (c) The average hourly wage will be $29.08 in 2025.