 # Representations of a Line

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Examples, solutions, worksheets, videos, and lessons to help Grade 8 students learn how to graph a line specified by a linear function.

### New York State Common Core Math Grade 8, Module 6, Lesson 3

Lesson 3 Student Outcomes

• Students graph a line specified by a linear function.
• Students graph a line specified by an initial value and rate of change of a function and construct the linear function by interpreting the graph.
• Students graph a line specified by two points of a linear relationship and provide the linear function.

Lesson 3 Summary

When the rate of change, b, and an initial value, a, are given in the context of a problem, the linear function that models the situation is given by the equation y=a + bx.

The rate of change and initial value can also be used to graph the linear function that models the situation.

When two or more ordered pairs are given in the context of a problem that involves a linear relationship, the graph of the linear function is the line that passes through those points. The linear function can be represented by the equation of that line.

Lesson 3 Classwork

Example 1: Rate of Change and Initial Value given in the Context of the Problem
A truck rental company charges a \$150 rental fee, in addition to a charge of \$0.50 per mile driven. In this problem, you will graph the linear function relating the total cost of the rental in dollars, C, to the number of miles driven, m, on the axes below.
a. If the truck is driven zero miles, what will be the cost to the customer? How will this be shown on the graph?
b. What is the rate of change that relates cost to number of miles driven? Explain what it means within the context of the problem.
c. On the axes given, graph the line that relates C to m.
d. Write the linear function that models the relationship between number of miles driven and total rental cost?

Exercises 1 - 5
Jenna bought a used car for \$18,000. She has been told that the value of the car is likely to decrease by \$2,500 for each year that she owns the car. Let the value of the car in dollars be V and the number of years Jenna has owned the car be t.
1. What is the value of the car when t = 0? Show this point on the graph.
2. What is the rate of change that relates V to t? (Hint: Is it positive or negative? How can you tell?)

3. Find the value of the car when
a. t = 1?
b. t = 2?
c. t = 7?

4. Plot the points for the values you found in Exercise 3, and draw the line (using a straight-edge) that passes through those points.

5. Write the linear function that models the relationship between the number of years Jenna has owned the car and the value of the car.

Exercises 6 - 10
An online bookseller has a new book in print. The company estimates that if the book is priced at \$15.00 then 800 copies of the book will be sold per day, and if the book is priced \$20 at then 550 copies of the book will be sold per day.
6. Identify the ordered pairs given in the problem. Then plot both on the graph.

7. Assume that the relationship between the number of books sold and the price is linear. (In other words, assume that the graph is a straight line.) Using a straight-edge, draw the line that passes through the two points.

8. What is the rate of change relating number of copies sold to price?

9. Based on the graph, if the company prices the book at \$18, about how many copies of the book can they expect to sell per day?

10. Based on the graph, approximately what price should the company charge in order to sell 700 copies of the book per day?

Lesson 3 Exit Ticket

1. A car starts a journey with 8 gallons of fuel. The car will consume 0.04 gallons for every mile driven. Let A represent the amount of gas in the tank (in gallons) and m represent the number of miles driven.

a. How much gas is in the tank if 0 miles have been driven? How would this be represented on the axes above?

b. What is the rate of change that relates the amount of gas in the tank to the number of miles driven? Explain what it means within the context of the problem.

c. On the axes above, graph the line that relates A to m.

d. Write the linear function that models the relationship between the number of miles driven and the amount of gas in the tank.

2. Andrew works in a restaurant. The graph below shows the relationship between the amount Andrew earns and the number of hours he works.

a. If Andrew works for7 hours, approximately how much does he earn?

b. Estimate how long Andrew has to work in order to earn \$6a?

c. What is the rate of change of the function given by the graph? Interpret the value within the context of the problem.

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