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Increasing and Decreasing Functions Examples




 


Videos to help Grade 8 students learn how to describe qualitatively the functional relationship between two types of quantities by analyzing a graph.

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Common Core For Grade 8

New York State Common Core Math Grade 8, Module 6, Lesson 5.

Lesson 5 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 5 Summary

The graph of the function can be used to help describe in relationship between two quantities.

The slope of the line can provide useful information about the functional relationship between two quantities:

  • A function whose graph has a positive slope is said to be an increasing function.
  • A function whose graph has a negative slope is said to be a decreasing function.
  • A function whose graph has a zero slope is said to be a constant function.

Lesson 5 Classwork

Example 1: Nonlinear Functions in the Real World
Not all real world situations can be modeled by a linear function. There are times when a nonlinear function is needed to describe the relationship between two types of quantities. Compare the two scenarios:
a. Aleph is running at a constant rate on a flat paved road. The graph below represents the total distance he covers with respect to time.
b. Shannon is running on a rocky trail that is initially flat and then leads up a steep mountain. The graph below represents the total distance she covers with respect to time.

Exercises 1 - 2
1. In your own words, describe what is happening as Aleph is running during the following intervals of time.
a. 0 to 15 minutes
b. 15 to 30 minutes
c. 30 to 45 minutes
d. 45 to 60 minutes
2. Describe in your own words what is happening as Shannon is running during the following intervals of time.
a. 0 to 15 minutes
b. 15 to 30 minutes
c. 30 to 45 minutes
d. 45 to 60 minutes
Example 2: Increasing and Decreasing Functions
The rate of change of a function can provide useful information about the relationship between two quantities. A linear function has a constant rate of change. A nonlinear function has a variable rate of change.

Exercises 3 - 5
3. Different breeds of dogs have different growth rates. A large breed dog typically experiences a rapid growth rate from birth to age 6 months. At that point, the growth rate begins to slow down until the dog reaches full growth around 2 years of age.
a. Sketch a graph that represents the weight of a dog from birth to 2 years of age.
b. Is the function represented by the graph linear or nonlinear? Explain.
c. Is the function represented by the graph increasing or decreasing? Explain.

4. Nikka took her laptop to school and drained the battery while typing a research paper. When she returned home, Nikka connected her laptop to a power source and the battery recharged at a constant rate.
a. Sketch a graph that represents the battery charge with respect to time.
b. Is the function represented by the graph linear or nonlinear? Explain.
c. Is the function represented by the graph increasing or decreasing? Explain.

5. The long jump is a track and field event where an athlete attempts to leap as far as possible from a given point. Mike Powell of the United States set the long jump world record of 8.95 meters (29.4 feet) during the 1991 World Championships in Tokyo, Japan.
a. Sketch a graph that represents the path of a high school athlete attempting the long jump.
b. Is the function represented by the graph linear or nonlinear? Explain.
c. Is the function represented by the graph increasing or decreasing? Explain.



Example 3: Ferris Wheel
Lamar and his sister are riding a Ferris wheel at a state fair. Using their watches, they find that it take 8 seconds for the Ferris wheel to make a complete revolution. The graph below represents Lamar and his sister’s distance above the ground with respect to time.

Exercises 6–9
6. Use the graph from Example 3 to answer the following questions.
a. Is the function represented by the graph linear or nonlinear?
b. Where is the function increasing? What does this mean within the context of the problem?
c. Where is the function decreasing? What does this mean within the context of the problem?
7. How high above the ground is the platform for passengers to get onto the Ferris wheel? Explain your reasoning.
8. Based on the graph, how many revolutions does the Ferris wheel complete during the 40 second time interval? Explain your reasoning.
9. What is the diameter of the Ferris wheel? Explain your reasoning.
Lesson 5 Exit Ticket

Lamar and his sister continue to ride the Ferris wheel. The graph below represents Lamar and sister’s distance above the ground with respect to time during the next 40 seconds of their ride.

a. Name one interval where the function is increasing.

b. Name one interval where the function is decreasing.

c. Is the function linear or nonlinear? Explain.

d. What could be happening during the interval of time from to seconds?

e. Based on the graph, how many complete revolutions are made during this 40 second interval?



 

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