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Linear Functions and Proportionality




 

Video solutions to help Grade 8 students know that a function assigns to each input exactly one output.


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Common Core For Grade 8

New York State Common Core Math Module 5, Grade 8, Lesson 3


Lesson 3 Student Outcomes

• Students relate constant speed and proportional relationships to linear functions using information from a table.
• Students know that distance traveled is a function of the time spent traveling and that the total cost of an item is a function of how many items are purchased.

Lesson 3 Student Summary

Functions can be described by a rule in the form of y = mx + b, where m and b are constants.
Constant rates and proportional relationships can be described by a function, specifically a linear function where the rule is a linear equation.
Functions are described in terms of their inputs and outputs. For example, if the inputs are related to time and the output are distances traveled at given time intervals then we say that the distance traveled is a function of the time spent traveling.

Lesson 3 Classwork

Example 1
In the last lesson we looked at several tables of values that represented the inputs and outputs of functions. For example:

Example 2
Walter walks 8 miles in two hours. What is his average speed?

Example 3
Veronica runs at a constant speed. The distance she runs is a function of the time she spends running. The function has the table of values shown below.

Example 4
Water flows from a faucet at a constant rate. That is, the volume of water that flows out of the faucet is the same over any given time interval. If 7 gallons of water flow from the faucet every minutes, determine the rule that describes the volume function of the faucet.
Now assume that you are filling the same tub, a tub with a volume of 50 gallons, with the same faucet, a faucet where the rate of water flow is 3.5 gallons per minute. This time, however, the tub already has 8 gallons in it. Will it still take 14 minutes to fill the tub? Explain.

Example 5
Water flows from a faucet at a constant rate. Assume that 6 gallons of water are already in a tub by the time we notice the faucet is on. This information is recorded as 0 minutes and 6 gallons of water in the table below. The other values show how many gallons of water are in the tub at the given number of minutes.




Exercises 1–3
1. A linear function has the table of values below. The information in the table shows the function of time in minutes with respect to mowing an area of lawn in square feet.
a. Explain why this is a linear function.
b. Describe the function in terms of area mowed and time.
c. What is the rate of mowing a lawn in 5 minutes?
d. What is the rate of mowing a lawn in 20 minutes?
e. What is the rate for mowing a lawn in 30 minutes?
f. What is the rate for mowing a lawn in 50 minutes?
g. Write the rule that represents the linear function that describes the area in square feet mowed, y, in x minutes.
h. Describe the limitations of x and y.
i. What number does the function assign to 24? That is, what area of lawn can be mowed in 24 minutes?
j. How many minutes would it take to mow an area of 400 square feet?

2. A linear function has the table of values below. The information in the table shows the volume of water that flows from a hose in gallons as a function of time in minutes.
a. Describe the function in terms of volume and time.
b. Write the rule that represents the linear function that describes the volume of water in gallons, y, in x minutes.
c. What number does the function assign to 250? That is, how many gallons of water flow from the hose in 250 minutes?
d. The average pool has about 17,300 gallons of water. The pool has already been filled 1/4 of its volume. Write the rule that describes the volume of water flow as a function of time for filling the pool using the hose, including the number of gallons that are already in the pool.
e. Approximately how much time, in hours, will it take to finish filling the pool?

3. Recall that a linear function can be described by a rule in the form of y = mx + b, where m and b are constants. A particular linear function has the table of values below.
a. What is the equation that describes the function?
b. Complete the table using the rule.


 

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