$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

2.1E: Linear Functions (Exercises)

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Section 2.1 exercise

1. A town's population has been growing linearly. In 2003, the population was 45,000, and the population has been growing by 1700 people each year. Write an equation,

$$P(t)$$, for the population $$t$$ years after 2003.

2. A town's population has been growing linearly. In 2005, the population was 69,000, and the population has been growing by 2500 people each year. Write an equation,

$$P(t)$$, for the population $$t$$ years after 2005.

3. Sonya is currently 10 miles from home, and is walking further away at 2 miles per hour. Write an equation for her distance from home $$t$$ hours from now.

4. A boat is 100 miles away from the marina, sailing directly towards it at 10 miles per hour. Write an equation for the distance of the boat from the marina after $$t$$ hours.

5. Timmy goes to the fair with $40. Each ride costs$2. How much money will he have left after riding n rides?

6. At noon, a barista notices she has $20 in her tip jar. If she makes an average of$0.50 from each customer, how much will she have in her tip jar if she serves $$n$$ more customers during her shift?

Determine if each function is increasing or decreasing

7. $$f(x) = 4x + 3$$

8. $$g(x) = 5x + 6$$

9. $$a(x) = -2x + 4$$

10. $$b(x) = 8 - 3x$$

11. $$h(x) = -2x + 4$$

12. $$h(x) = -4x + 1$$

13. $$j(x) = \dfrac{1}{2}x - 3$$

14. $$p(x) = \dfrac{1}{4} x - 5$$

15. $$n(x) = -\dfrac{1}{3} x - 2$$

16. $$m(x) = -\dfrac{3}{8} x + 3$$

Find the slope of the line that passes through the two given points

17. (2, 4) and (4, 10)

18. (1, 5) and (4, 11)

19. (-1, 4) and (5, 2)

20. (-2, 8) and (4, 6)

21. (6, 11) and (-4, 3)

22. (9, 10) and (-6, -12)

Find the slope of the lines graphed

23. 24. 25. Sonya is walking home from a friend’s house. After 2 minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home. What is her rate?

26. A gym membership with two personal training sessions costs $125, while gym membership with 5 personal training sessions costs$260. What is the rate for personal training sessions?

27. A city's population in the year 1960 was 287,500. In 1989 the population was 275,900. Compute the slope of the population growth (or decline) and make a statement about the population rate of change in people per year.

28. A city's population in the year 1958 was 2,113,000. In 1991 the population was 2,099,800. Compute the slope of the population growth (or decline) and make a statement about the population rate of change in people per year.

29. A phone company charges for service according to the formula: $$C(n) = 24 0.+1n$$, where $$n$$ is the number of minutes talked, and $$C(n)$$ is the monthly charge, in dollars. Find and interpret the rate of change and initial value.

30. A phone company charges for service according to the formula: $$C(n) = 26 0.+04n$$, where $$n$$ is the number of minutes talked, and $$C(n)$$ is the monthly charge, in dollars. Find and interpret the rate of change and initial value.

31. Terry is skiing down a steep hill. Terry's elevation, $$E(t)$$, in feet after $$t$$ seconds is given by $$E(t) = 3000 - 70t$$.Write a complete sentence describing Terry’s starting elevation and how it is changing over time.

32. 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.

Given each set of information, find a linear equation satisfying the conditions, if possible

33. $$f(-5) = -4$$, and $$f(5) = 2$$

34. $$f(-1) = 4$$, and $$f(5) = 1$$

35. Passes through (2, 4) and (4, 10)

36. Passes through (1, 5) and (4, 11)

37. Passes through (-1, 4) and (5, 2)

38. Passes through (-2, 8) and (4, 6)

39. $$x$$ intercept at (-2, 0) and $$y$$ intercept at (0, -3)

40. $$x$$ intercept at (-5, 00 and $$y$$ intercept at (0, 4)

Find an equation for the function graphed

41. 42. 43. 44. 45. A clothing business finds there is a linear relationship between the number of shirts, $$n$$, it can sell and the price, $$p$$, it can charge per shirt. In particular, historical data shows that 1000 shirts can be sold at a price of $30, while 3000 shirts can be sold at a price of$22 . Find a linear equation in the form $$p = mn + b$$ that gives the price $$p$$ they can charge for $$n$$ shirts.

46. A farmer finds there is a linear relationship between the number of bean stalks, $$n$$, she plants and the yield, $$y$$, each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationships in the form $$y = mn + b$$ that gives the yield when $$n$$ stalks are planted.

47. Which of the following tables could represent a linear function? For each that could be linear, find a linear equation models the data. 48. Which of the following tables could represent a linear function? For each that could be linear, find a linear equation models the data. 49. While speaking on the phone to a friend in Oslo, Norway, you learned that the current temperature there was -23 Celsius (-23$${}^{o}$$C). After the phone conversation, you wanted to convert this temperature to Fahrenheit degrees, oF, but you could not find a reference with the correct formulas. You then remembered that the relationship between $${}^{o}$$F and $${}^{o}$$C is linear. [UW]

Using this and the knowledge that 32$${}^{o}$$F = 0 $${}^{o}$$C and 212 $${}^{o}$$F = 100 $${}^{o}$$C, find an equation that computes Celsius temperature in terms of Fahrenheit; i.e. an equation of the form C = “an expression involving only the variable F.”

Likewise, find an equation that computes Fahrenheit temperature in terms of Celsius temperature; i.e. an equation of the form F = “an expression involving only the variable C.”

How cold was it in Oslo in $${}^{o}$$F?

1. $$P(t) = 1700t + 45000$$

3. $$D(t) = 10 + 2t$$

5. $$M(n) = 40 - 2n$$

7. Increasing

9. Decreasing

11. Decreasing

13. Increasing

15. Decreasing

17. 3

19. $$-\dfrac{1}{3}$$

21. $$\dfrac{4}{3}$$

23. $$\dfrac{2}{3}$$

25. -0.05 mph (or 0.05 miles per hour toward her home)

27. Population is decreasing by 400 people per year

29. Monthly charge in dollars has an initial base charge of $24, and increases by$0.10 for each minute talked

31. Terry started at an elevation of 3,000 ft and is descending by 70ft per second.

33. $$y = \dfrac{3}{5} x - 1$$

35. $$y = 3x - 2$$

37. $$y = -\dfrac{1}{3}x + \dfrac{11}{3}$$

39. $$y = -1.5x - 3$$

41. $$y = \dfrac{2}{3} x + 1$$

43. $$y = -2x + 3$$

45. $$P(n) = -0.004n + 34$$

47. The $$1^{\text{st}}$$, $$3^{\text{rd}}$$ & $$4^{\text{th}}$$ tables are linear: respectively
1. $$g(x) = -3x + 5$$
3. $$f(x) = 5x - 5$$
4. $$k(x) = 3x - 2$$

49a. $$C = \dfrac{5}{9} F - \dfrac{160}{9}$$
b. $$F = \dfrac{9}{5} C + 32$$
c. $$-9.4^{\cric} F$$