Interpreting Rate of Change and Initial Value

Examples, solutions, videos, and lessons to help Grade 8 students learn how to interpret the rate of change and initial value of a line in context.

New York State Common Core Math Grade 8, Module 6, Lesson 2

Worksheets for Grade 8

Lesson 2 Student Outcomes

• Students interpret the rate of change and initial value of a line in context.
• Students interpret slope as rate of change and relate slope to the steepness of a line and the sign of the slope, indicating that a linear function is increasing if the slope is positive and decreasing if the slope is negative.

Lesson 2 Summary

When a linear function is given by the equation of a line of the form y = mx +c, the rate of change is m and initial value is b. Both are easy to identify.

The rate of change of a linear function is the slope of the line it represents. It is the change in the values of y per a one unit increase in the values of x.

A positive rate of change indicates that a linear function is increasing.
A negative rate of change indicates that a linear function is decreasing.
Given two lines each with positive slope, the function represented by the steeper line has a greater rate of change.

The initial value of a linear function is the value of the y-variable when the x value is zero.

Lesson 2 Classwork

Linear functions are defined by the equation of a line. The graphs and the equations of the lines are important for understanding the relationship between the two variables represented in the following example as x and y.

Example 1: Rate of Change and Initial Value
The equation of a line can be interpreted as defining a linear function. The graphs and the equations of lines are important in understanding the relationship between two types of quantities (represented in the following examples by x and y).
In a previous lesson, you encountered an MP3 download site that offers downloads of individual songs with the following price structure: a $3 fixed fee for monthly subscription PLUS a fee of $0.25 per song. The linear function that models the relationship between the number of songs downloaded and the total monthly cost of downloading songs can be written as y = 0.25x + 3, where x represents the number of songs downloaded and y represents the total monthly cost (in dollars) for MP3 downloads.
a. In your own words, explain the meaning of 0.25 within the context of the problem.
b. In your own words, explain the meaning of 3 within the context of the problem.

Exercises 1 - 6
Another site offers MP3 downloads with a different price structure: a $2 fixed fee for monthly subscription PLUS a fee of $0.40 per song.
1. Write a linear function to model the relationship between the number of songs download and the total monthly cost. As before, let x represent the number of songs downloaded and y represent the total monthly cost (in dollars) of downloading songs.

2. Determine the cost of downloading 0 songs and 10 songs from this site.

3. The graph below already shows the linear model for the first subscription site (Company 1): y = 0.25x + 3. Graph the equation of the line for the second subscription site (Company 2) by marking the two points from your work above (for 0 songs and 10 songs) and drawing a line through those two points.

4. Which line has a steeper slope? Which company's model has the more expensive cost per song?

5. Which function has the greater initial value?

6. Which subscription site would you choose if you only wanted to download 5 songs per month? Which company would you choose if you wanted to download 10 songs? Explain your reasoning.

Exercises 7 - 9
7. When someone purchases a new car and begins to drive it, the mileage (meaning the number of miles the car has traveled) immediately increases. Let represent the number of years since the car was purchased and represent the total miles traveled. The linear function that models the relationship between the number of years since purchase and the total miles traveled is given by: y = 15,000x.
a. Identify and interpret the rate of change.
b. Identify and interpret the initial value.
c. Is the mileage increasing or decreasing each year according to the model? Explain your reasoning.

8. When someone purchases a new car and begins to drive it, generally speaking, the resale value of the car (in dollars) goes down each year. Let x represent the number of years since purchase and y represent the resale value of the car (in dollars). The linear function that models the resale value based on the number of years since purchase is given by: y = 20,000 - 1,200x
a. Identify and interpret the rate of change.
b. Identify and interpret the initial value.
c. Is the resale value increasing or decreasing each year according to the model? Explain.

9. Suppose you are given the linear function y = 2.5x + 10.
a. Write a story that can be modeled by the given linear function.
b. What is the rate of change? Explain its meaning with respect to your story.
c. What is the initial value? Explain its meaning with respect to your story.

Lesson 2 Exit Ticket

In 2008, a collector of sports memorabilia purchased 5 specific baseball cards as an investment. Let y represent the card’s resale value (in dollars) and x represent the number of years since purchase. Each of the cards' resale values after 0, 1, 2, 3, and 4 years could be modeled by linear equations as follows:

Card A: y = 5 - 0.7x

Card B: y = 4 + 2.6x

Card C: y = 10 + 0.9x

Card D: y = 10 - 1.1x

Card E: y = 8 + 0.25x

1. Which card(s) are decreasing in value each year? How can you tell?

2. Which card(s) had the greatest initial values at purchase (at 0 years)?

3. Which card(s) is increasing in value the fastest from year to year? How can you tell?

4. If you were to graph the equations of the resale values of Card B and Card C, which card's graph line would be steeper? Explain.

5. Write a sentence explaining the “0.9" value in the “Card C” equation.

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