Interpreting the y-intercept and Slope



4E Interpreting the y-intercept and Slope

y-intercept: The amount of y (dependent variable) at the beginning when x. It’s always Y unit when x = 0

(Independent variable) is 0

Slope: The change of y (dependent variable) for every x. The unit is always Y units per X unit

Equation: y = slope x + y-intercept.

Example 1: A taxi charges $.50 per 1/3 mile plus a flat fee of $4.

Slope: .50/(1/3) = $1.50 per mile y-intercept: 4

Equation: y = 1.50x + 4

Example 2: The following table shows the number of thingamijigs based on the number of whatamacallits. What is the meaning of the slope and the meaning of the y-intercept?

|Whatamacallits |3 |6 |9 |

|Thingamijgs |8 |16 |24 |

Slope: Number of Thingamijigs per Whatamacallits

Y-intercept: Number of Thingamajigs at 0 Whatamacallits

Example 3: Jackson bought a computer that came with 30 programs. He added five programs to his computer every month.

a) How many programs were at the beginning (y-intercept)? 30

b) How many programs are increasing each month (slope)? 5

c) The interpretation of the y-intercept is: 30 programs at 0 months.

d) The interpretation of the slope: Programs are increasing by 5 per month.

e) The equation: P = 5m + 30

Example 4: Write an example involving stairs that has a slope of 2 and a y-intercept of 4. Interpret the slope and y-intercept.

John exited the elevator on the 4th floor of a building and went to the stairs. He climbed 2 floors each minute.

y-intercept: He was on the 4th floor at zero minutes. Slope: The amount of floors he climbed each minute was 2.

Practice Problems

1. A car rental company charges $2 for every 10 miles and a daily rate of $30

a. What is the slope per mile? b. What is the y-intercept? c. What is the equation?

2. A taxi charges $3 for every ½ mile and a flat fee of $2.

a. What is the rate of change per mile? b. What is the equation?

3. The following table shows the vertical distance and horizontal distance of a football thrown in the air:

|Horizontal Distance (ft) |30 |50 |70 |80 |

|Vertical Distance (ft) |12 |20 |28 |32 |

a. Interpret the slope: b. Interpret the y-intercept:

4. Julie recently purchased a Smartphone with 50 songs. She could download on average about 5 songs a week.

a. What is the y-intercept (how many are at the beginning)?

b. Interpret the y-intercept (what does the beginning represent?

c. What is the slope (what is the rate of change?)

d. Interpret the slope (what is changing and use “per”)

e. Write an equation.

5. Mrs. Hudson, a Social Studies teacher, noticed that her students knew about 10 state capitols before she started teaching the unit on State Capitols. For every half-hour that the students studied, they learned 8 more capitols.

a. What is the y-intercept? b. Interpret the y-intercept

c. What is the slope? d. Interpret the slope?

e. What is the equation?

6. Delaney filled her car’s gas tank with 17 gallons. The amount of gas decreased by 1 gallon for every 25 miles she drove.

a. What is the y-intercept? b. Interpret the y-intercept?

c. What is the slope? d. Interpret the slope?

e. What is the equation?

7. Write an example that has a y-intercept of 15 green beans and a slope of 10 green beans per scoop. Then interpret y-

intercept and slope.

8. Write an example that has a y-intercept of $40 and a slope of $.15 per text after 200 texts

9. Create an example that has a y-intercept and a slope. Then write the y-intercept, interpretation of the y-intercept, slope,

and interpretation of the slope. Ideas include: Taxi-ride fare; filling a swimming pool; words per minute that someone can type with lessons etc…

10. If the slope is .0056 gallons per mile then what is the slope per 100 miles?

11. The function below shows the cost of a hamburger with different numbers of toppings (t). f(t) = 1.90 + 1.40t

a. What is the y-intercept, and what does it mean?

b. What is the slope, and what does it mean?

c. If Jodi paid $3.30 for a hamburger, how many toppings were on Jodi’s hamburger?

12. The function below shows the cost of an ice cream sundae with different numbers of toppings (t). f(t) = 2.25 + 0.75t

a. What is the y-intercept, and what does it mean?

b. What is the slope, and what does it mean?

c. If Kaye paid $6.00 for a sundae, how many toppings were on Kaye’s sundae?

13. The function below shows the cost to attend the fair if you ride r rides. f(r) = 5 + 1.75r

a. What is the y-intercept, and what does it mean?

b. What is the slope, and what does it mean?

c. If Al spent $19.00 at the fair, how many rides did Al ride?

14. The function below shows the cost for Mrs. Franklin to go to a buffet with c of her grandchildren. f(c) = 6.85 + 2.95c

a. What is the y-intercept, and what does it mean?

b. What is the slope, and what does it mean?

c. If Mrs. Franklin paid 18.65 for the buffet, how many of her grandchildren did she take to the buffet?

15. The graph below shows the number of newspapers delivered and total pay for Leona’ newspaper delivery job. What does the slope of this graph represent?

16. Colby put $100 in a savings account. The graph to the right shows how the amount in the account would increase over the next ten years. What does the y-intercept represent?

|Thingathings (x) |0 |8 |16 |

|Falties (y) |7 |12 |17 |

17. a. What is the slope and what does it represent?

b. What is the y-intercept and what does it represent

18. Dionne pays a fixed fee plus an hourly rate to rent a boat. The table below shows how much Dionne paid for the boat. What was Dionne’s hourly rate to rent the boat?

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19. Rich is a member of a gym. He pays a monthly fee plus a per-visit fee. The equation below represents the monthly amount Rich pays for his membership to the gym per month for x visits. y = 3x + 10. What does the y-intercept of the graph of this equation represent?

20. Nan works as a commissioned sale rep. She makes a weekly base salary plus a commission for each sale she makes. The table below shows how much Nan can make. What is Nan’s weekly base salary?

21. Charlie rented a moving truck. He paid a daily fee plus a per-mile fee to rent the truck. The equation below represents the daily amount Charlie paid for the truck if he drives it x miles. y = 0.5x + 10 What does the slope of the graph of this equation represent?

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