Activities

1. Determine the slope of the following equations:
   1. y = 6x – 5
   2. y = 4 – 3x
   3. y = -2
   4. y = 3(4x – 1)
   5. 5y = -6x

2. What is the y-intercept of the following equations:
   1. y = 2x + 3
   2. y = -1 – 3x
   3. y = 0
   4. y = 4(x – ½)
   5. 7y = -11x

3. Given the following points, (x₁, y₁) and (x₂, y₂) what expression would give you the slope of the line connecting the points?

4. In the equation y = mx + b, write a short description, in your own words, of what the y, m, x, and b represent.

5. Assuming b is the y-intercept of a linear equation that includes the following point, what goes in the blank?                   

(___, b)

6. What is the equation of the line, in slope intercept form, graphed here:

7. Given the equation 3y + 2 = x – 5, what is the slope and y-intercept of this line?

8. The linear equation c = 8 + 1.75t is used to determine the cost of a pizza with t toppings. What is the slope and y-intercept of this equation?

9. The number of people who lived in a certain city grew by 1,200 each year after 1980. If there were 51,200 people in the city by 1981, what is the linear equation that describes the town growth since 1980?

10. What is the slope and y-intercept of the graph here?

11. An extreme winter storm caused the temperature to plummet according to the graph below. What is the slope and y-intercept of the graph?

12. What is the slope and y-intercept of the equation y = 0?

13. Jonathan just got his first job since graduating college. He wants to spend his money wisely so he has set up a monthly budget. Jonathan determines that he will make $1,314.56 dollars per month after taxes. Considering all of his expenses he will spend $989.12 per month and will put the rest towards his college debt. If Jonathan was $24,117.22 in debt before starting the job, write an equation that describes his debt remaining (d) based on the months (m) since he started his new job.

14. When buying specialty lumber, Luke, a woodworker needs to calculate the number of board feet in the material and multiply by the cost per board foot to determine the total cost of the piece of lumber. The formula to calculate the cost for a piece of lumber is:

Luke wants to simplify the function so he can calculate the costs of three species of wood that all have the same dimensions. If the lumber is 1 inch thick, by 8 inches wide, by 96 inches long, what is the slope of the simplified function?

15. During a basketball game the Dragons made 12 three pointers and 11 foul shots (1 point each) and the rest of their points came from two point baskets. Determine the slope and y-intercept for a function that outputs the total points the Dragons scored (p) based on the number of 2-point baskets made (b).

16.  Given the equation , create an equation that is parallel and an equation that is perpendicular to the original line.

17. Andy’s rainwater barrel drains water onto his garden according to the graph below where the independent variable is the time spent draining, in minutes, and the dependent variable is amount of water still in the barrel, in gallons. What are the values of the slope and y-intercept of this graph and what do they represent in terms of the rain barrel?

18. Compare and contrast the slope intercept form of a linear equation (y = mx + b), with the point slope form of an equation (y – y₁ = m(x – x₁))

19. What could be the equation of the following linear function?

20. Why is the slope of a vertical line not defined?

21. If the slope of a linear function is m = 6/5, how much does the output change, when the input is decreased by 30?

22. Jackson just got a new job so that he could buy NBA trading cards. Four months after he began working he had a total of 31 trading cards. Nine months after he began working he has a total of 46 trading cards. Assuming he bought the same amount of trading cards per month, how many trading cards did he have before he got his job?

23. In the linear equation cx – dy = e, assuming x is the input and y is the output, find the values for the slope and starting point of this equation.

24. Given the equation what is the slope and y-intercept.

25. Jeremy is trying to decrease his 5k race time by 1 minute per month. He most recently ran a 5k in 29:32. Write an equation that would determine his new 5k race time (t) based on the number of months (m) since running the 29:32. Explain why this function could not continue indefinitely with the same slope.

Answer Key/Rubric

1. Determine the slope of the following equations:
   1. 6
   2. -3
   3. 0
   4. 12
   5. -6/5

2. What is the y-intercept of the following equations:
   1. 3
   2. -1
   3. 0
   4. -2
   5. 0

4. y is the output value, m is the slope (or rate of change), x is the input value, and b is the starting point (or y-intercept)

5. (0, b)

6. 
   
   
7. ; slope = 1/3, y-intercept = -7/3

8. The slope is 1.75 because that is the rate that the cost changes for each additional topping. The y-intercept is 8 because that is the starting value for a pizza with 0 toppings.

9. y = 1200x + 50,000

10. ; slope = -1/4, y-intercept = 1

11. m = -(4/3), b = 20

12. m = 0, b = 0

13. d = 24,117.22 – (1314.56 – 989.12)m

14. m = 16/3

15. p = 2b + 47

16. Various answers including: Parallel: Perpendicular

17. m = 5/4 gallons per minute, b = 40 gallons. This means that Andy’s rainwater barrel holds a total of 40 gallons of water and when draining, 5 gallons are removed every 4 minutes.

18. Point slope and slope intercept form are two versions of a similar thing. For instance, when using slope-intercept form, a student would start with determining the slope using the slope formula. That is the same as the point slope form, if it were divided by (x – x₁) on both sides of the equation. Additionally, to find the y-intercept of a point slope equation you would multiply m times (-x₁) then add y₁. This is the same as what happens when you plug a point (x, y) and the slope into the y=mx + b equation and solve for b.

19. Various answers including:

20. A vertical line is not considered to have a slope because when you look at the change in y divided by the change in x, you notice that there is no change in x. Thus the fraction has a denominator of zero and cannot be divided.

21. The output would be decreased by 36

22. He had 19 cards

23.  , the slope is (c/d) and the y-intercept is (-e/d)

24. m = (5/6), b = -(51/6)

25. t = 29:32 – m , this function could not continue in this way indefinitely because at some point, once Jeremy gets so much better, he will not be able to run one minute faster each month. In other words, he cannot continue to improve at this rate or he will eventually be running five kilometers in a negative amount of time.