4.2 Modeling with linear functions  (Page 6/9)

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Access this online resource for additional instruction and practice with linear function models.

Key concepts

• We can use the same problem strategies that we would use for any type of function.
• When modeling and solving a problem, identify the variables and look for key values, including the slope and y -intercept. See [link] .
• Check for reasonableness of the answer.
• Linear models may be built by identifying or calculating the slope and using the y -intercept.
• The x -intercept may be found by setting $\text{\hspace{0.17em}}y=0,$ which is setting the expression $\text{\hspace{0.17em}}mx+b\text{\hspace{0.17em}}$ equal to 0.
• The point of intersection of a system of linear equations is the point where the x - and y -values are the same. See [link] .
• A graph of the system may be used to identify the points where one line falls below (or above) the other line.

Verbal

Explain how to find the input variable in a word problem that uses a linear function.

Determine the independent variable. This is the variable upon which the output depends.

Explain how to find the output variable in a word problem that uses a linear function.

Explain how to interpret the initial value in a word problem that uses a linear function.

To determine the initial value, find the output when the input is equal to zero.

Explain how to determine the slope in a word problem that uses a linear function.

Algebraic

Find the area of a parallelogram bounded by the y -axis, the line $\text{\hspace{0.17em}}x=3,$ the line $\text{\hspace{0.17em}}f\left(x\right)=1+2x,$ and the line parallel to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ passing through $\text{\hspace{0.17em}}\left(\text{2},\text{7}\right).$

6 square units

Find the area of a triangle bounded by the x -axis, the line $\text{\hspace{0.17em}}f\left(x\right)=12–\frac{1}{3}x,$ and the line perpendicular to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ that passes through the origin.

Find the area of a triangle bounded by the y -axis, the line $\text{\hspace{0.17em}}f\left(x\right)=9–\frac{6}{7}x,$ and the line perpendicular to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ that passes through the origin.

20.01 square units

Find the area of a parallelogram bounded by the x -axis, the line $\text{\hspace{0.17em}}g\left(x\right)=2,$ the line $\text{\hspace{0.17em}}f\left(x\right)=3x,$ and the line parallel to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ passing through $\text{\hspace{0.17em}}\left(6,1\right).$

For the following exercises, consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped 4,700. Assume this trend continues.

Predict the population in 2016.

2,300

Identify the year in which the population will reach 0.

For the following exercises, consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.

Predict the population in 2016.

64,170

Identify the year in which the population will reach 75,000.

For the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.

Find the linear function that models the town’s population $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ as a function of the year, $\text{\hspace{0.17em}}t,$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is the number of years since the model began.

$P\left(t\right)=75,000+2500t$

Find a reasonable domain and range for the function $\text{\hspace{0.17em}}P.$

If the function $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ is graphed, find and interpret the x - and y -intercepts.

(–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000.

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