Lesson 8Linear Functions
Learning Goal
Let’s investigate linear functions.
Learning Targets
I can determine whether a function is increasing or decreasing based on whether its rate of change is positive or negative.
I can explain in my own words how the graph of a linear function relates to its rate of change and initial value.
Warm Up: Bigger and Smaller
Problem 1
Diego said that these graphs are ordered from smallest to largest. Mai said they are ordered from largest to smallest. But these are graphs, not numbers! What do you think Diego and Mai are thinking?
Activity 1: Proportional Relationships Define Linear Functions
Problem 1
Jada earns $7 per hour mowing her neighbors’ lawns.
Name the two quantities in this situation that are in a functional relationship. Which did you choose to be the independent variable? What is the variable that depends on it?
Write an equation that represents the function.
Here is a graph of the function. Label the axes. Label at least two points with input-output pairs.
Problem 2
To convert feet to yards, you multiply the number of feet by
Name the two quantities in this situation that are in a functional relationship. Which did you choose to be the independent variable? What is the variable that depends on it?
Write an equation that represents the function.
Draw the graph of the function. Label at least two points with input-output pairs.
Activity 2: Is It Filling Up or Draining Out?
Problem 1
There are four tanks of water.
The amount of water in gallons,
, in Tank A is given by the function , where is in minutes. The amount of water in gallons,
, in Tank B starts at 400 gallons and is decreasing at 5 gallons per minute. These functions work when and .
Which tank started out with more water?
Write an equation representing the relationship between
and . One tank is filling up. The other is draining out. Which is which? How can you tell?
The amount of water in gallons,
, in Tank C is given by the function . Is it filling up or draining out? Can you tell just by looking at the equation? The graph of the function for the amount of water in gallons,
, in Tank D at time is shown. Is it filling up or draining out? How do you know?
Are you ready for more?
Problem 1
Pick a tank that was draining out. How long did it take for that tank to drain? What percent full was the tank when
of that time had elapsed? When of the time had elapsed? What point in the plane is
of the way from to ? of the way? What point in the plane is
of the way from to ? of the way?
Activity 3: Which Is Growing Faster?
Problem 1
Noah is depositing money in his account every week to save money. The graph shows the amount he has saved as a function of time since he opened his account.
Elena opened an account the same day as Noah. The amount of money
Who started out with more money in their account? Explain how you know.
Who is saving money at a faster rate? Explain how you know.
How much will Noah save over the course of a year if he does not make any withdrawals? How long will it take Elena to save that much?
Lesson Summary
Suppose a car is traveling at 30 miles per hour. The relationship between the time in hours and the distance in miles is a proportional relationship. We can represent this relationship with an equation of the form
More generally, if we represent a linear function with an equation like