 # Increasing and Decreasing Functions

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Examples, solutions, worksheets, videos, and lessons to help Grade 8 students learn how to describe qualitatively the functional relationship between two types of quantities by analyzing a graph.

### New York State Common Core Math Grade 8, Module 6, Lesson 4

Lesson 4 Student Outcomes

• Students describe qualitatively the functional relationship between two types of quantities by analyzing a graph.
• Students sketch a graph that exhibits the qualitative features of a function based on a verbal description.

Lesson 4 Classwork

Graphs are useful tools in terms of representing data. They provide a visual story, highlighting important facts that surround the relationship between quantities.
The graph of a linear function is a line. The slope of the line can provide useful information about the functional relationship between the two types of quantities:
• A linear function whose graph has a positive slope is said to be an increasing function.
• A linear function whose graph has a negative slope is said to be a decreasing function.
• A linear function whose graph has a zero slope is said to be a constant function.

Exercises
1. Read through each of the scenarios and choose the graph of the function that best matches the situation. Explain the reason behind each choice.
a. A bathtub is filled at a constant rate of gallons per minute.
b. A bathtub is drained at a constant rate of gallons per minute.
c. A bathtub contains gallons of water.
d. A bathtub is filled at a constant rate of gallons per minute.

2. Read through each of the scenarios and sketch a graph of a function that models the situation.
a. A messenger service charges a flat rate of \$4.95 to deliver a package regardless of distance to the destination.
b. At sea level, the air that surrounds us presses down on our bodies at 14.7 pounds per square inch (psi). For every meters that you dive under water, the pressure increases by 14.7 psi.
c. The range (driving distance per charge) of an electric car varies based on the average speed the car is driven. The initial range of the electric car after a full charge is 400 miles. However, the range is reduced by 20 miles for every 10 mph increase in average speed the car is driven.

3. The graph below represents the total number of smart phones that are shipped to a retail store over the course of 50 days.

Match each part of the graph (A–C) to its verbal description. Explain the reasoning behind your choice.
i. Half of the factory workers went on strike and not enough smartphones were produced for normal shipments.
ii. The production schedule was normal and smartphones were shipped to the retail store at a constant rate.
iii. A defective electronic chip was found and the factory had to shut down, so no smartphones were shipped.

4. The relationship between Jameson’s account balance and time is modeled by the graph below.
a. Write a story that models the situation represented by the graph.
b. When is the function represented by the graph increasing? How does this relate to your story?
c. When is the function represented by the graph decreasing? How does this relate to your story?

Lesson 4 Exit Ticket

1. The graph below shows the relationship between a car’s value and time.
Match each part of the graph (A–C) to its verbal description. Explain the reasoning behind your choice.

i. The value of the car holds steady due to a positive consumer report on the same model.

ii. There is a shortage of used cars on the market and the value of the car rises at a constant rate.

iii. The value of the car depreciates at a constant rate.

2. Henry and Roxy both drive electric cars that need to be recharged before use. Henry uses a standard charger at his home to recharge his car. The graph below represents the relationship between the battery charge and the amount of time it has been connected to the power source for Henry’s car.

a. Describe how Henry’s car battery is being recharged with respect to time.

b. Roxy has a supercharger at her home that can charge about half of the battery in 20minutes. There is no remaining charge left when she begins recharging the battery. Sketch a graph that represents the relationship between the battery charge and the amount of time on the axes above.

c. Which person’s car will be recharged to full capacity first? Explain.

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