# 4.1 Linear functions  (Page 16/27)

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Write an equation for a line perpendicular to $\text{\hspace{0.17em}}h\left(t\right)=-2t+4\text{\hspace{0.17em}}$ and passing through the point $\text{\hspace{0.17em}}\left(-4,–1\right).$

Write an equation for a line perpendicular to $\text{\hspace{0.17em}}p\left(t\right)=3t+4\text{\hspace{0.17em}}$ and passing through the point $\text{\hspace{0.17em}}\left(3,1\right).$

$y=-\frac{1}{3}t+2$

## Graphical

For the following exercises, find the slope of the line graphed.

0

For the following exercises, write an equation for the line graphed.

$y=-\frac{5}{4}x+5$

$y=3x-1$

$y=-2.5$

For the following exercises, match the given linear equation with its graph in [link] .

$f\left(x\right)=-x-1$

$f\left(x\right)=-2x-1$

F

$f\left(x\right)=-\frac{1}{2}x-1$

$f\left(x\right)=2$

C

$f\left(x\right)=2+x$

$f\left(x\right)=3x+2$

A

For the following exercises, sketch a line with the given features.

An x -intercept of $\text{\hspace{0.17em}}\left(–4,\text{0}\right)\text{\hspace{0.17em}}$ and y -intercept of $\text{\hspace{0.17em}}\left(0,\text{–2}\right)$

An x -intercept $\text{\hspace{0.17em}}\left(–2,\text{0}\right)\text{\hspace{0.17em}}$ and y -intercept of $\text{\hspace{0.17em}}\left(0,\text{4}\right)$

A y -intercept of $\text{\hspace{0.17em}}\left(0,\text{7}\right)\text{\hspace{0.17em}}$ and slope $\text{\hspace{0.17em}}-\frac{3}{2}$

A y -intercept of $\text{\hspace{0.17em}}\left(0,\text{3}\right)\text{\hspace{0.17em}}$ and slope $\text{\hspace{0.17em}}\frac{2}{5}$

Passing through the points $\text{\hspace{0.17em}}\left(–6,\text{–2}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(6,\text{–6}\right)$

Passing through the points $\text{\hspace{0.17em}}\left(–3,\text{–4}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,\text{0}\right)$

For the following exercises, sketch the graph of each equation.

$f\left(x\right)=-2x-1$

$f\left(x\right)=-3x+2$

$f\left(x\right)=\frac{1}{3}x+2$

$f\left(x\right)=\frac{2}{3}x-3$

$f\left(t\right)=3+2t$

$p\left(t\right)=-2+3t$

$x=3$

$x=-2$

$r\left(x\right)=4$

For the following exercises, write the equation of the line shown in the graph.

$y=\text{3}$

$x=-3$

## Numeric

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

 $x$ 0 5 10 15 $g\left(x\right)$ 5 –10 –25 –40

Linear, $\text{\hspace{0.17em}}g\left(x\right)=-3x+5$

 $x$ 0 5 10 15 $h\left(x\right)$ 5 30 105 230
 $x$ 0 5 10 15 $f\left(x\right)$ –5 20 45 70

Linear, $\text{\hspace{0.17em}}f\left(x\right)=5x-5$

 $x$ 5 10 20 25 $k\left(x\right)$ 13 28 58 73
 $x$ 0 2 4 6 $g\left(x\right)$ 6 –19 –44 –69

Linear, $\text{\hspace{0.17em}}g\left(x\right)=-\frac{25}{2}x+6$

 $x$ 2 4 8 10 $h\left(x\right)$ 13 23 43 53
 $x$ 2 4 6 8 $f\left(x\right)$ –4 16 36 56

Linear, $\text{\hspace{0.17em}}f\left(x\right)=10x-24$

 $x$ 0 2 6 8 $k\left(x\right)$ 6 31 106 231

## Technology

For the following exercises, use a calculator or graphing technology to complete the task.

If $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is a linear function, find an equation for the function.

$f\left(x\right)=-58x+17.3$

Graph the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ on a domain of $\text{\hspace{0.17em}}\left[–10,10\right]:f\left(x\right)=0.02x-0.01.\text{\hspace{0.17em}}$ Enter the function in a graphing utility. For the viewing window, set the minimum value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to be $\text{\hspace{0.17em}}-10\text{\hspace{0.17em}}$ and the maximum value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to be $\text{\hspace{0.17em}}10.$

Graph the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ on a domain of $\text{\hspace{0.17em}}\left[–10,10\right]:fx\right)=2,500x+4,000$

[link] shows the input, $\text{\hspace{0.17em}}w,$ and output, $\text{\hspace{0.17em}}k,$ for a linear function $\text{\hspace{0.17em}}k.\text{\hspace{0.17em}}$ a. Fill in the missing values of the table. b. Write the linear function $\text{\hspace{0.17em}}k,$ round to 3 decimal places.

 w –10 5.5 67.5 b k 30 –26 a –44

[link] shows the input, $\text{\hspace{0.17em}}p,$ and output, $\text{\hspace{0.17em}}q,$ for a linear function $\text{\hspace{0.17em}}q.\text{\hspace{0.17em}}$ a. Fill in the missing values of the table. b. Write the linear function $\text{\hspace{0.17em}}k.$

 p 0.5 0.8 12 b q 400 700 a 1,000,000

Graph the linear function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ on a domain of $\text{\hspace{0.17em}}\left[-10,10\right]\text{\hspace{0.17em}}$ for the function whose slope is $\text{\hspace{0.17em}}\frac{1}{8}\text{\hspace{0.17em}}$ and y -intercept is $\text{\hspace{0.17em}}\frac{31}{16}.\text{\hspace{0.17em}}$ Label the points for the input values of $\text{\hspace{0.17em}}-10\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}10.$

Graph the linear function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ on a domain of $\text{\hspace{0.17em}}\left[-0.1,0.1\right]\text{\hspace{0.17em}}$ for the function whose slope is 75 and y -intercept is $\text{\hspace{0.17em}}-22.5.\text{\hspace{0.17em}}$ Label the points for the input values of $\text{\hspace{0.17em}}-0.1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}0.1.$

Graph the linear function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}f\left(x\right)=ax+b\text{\hspace{0.17em}}$ on the same set of axes on a domain of $\text{\hspace{0.17em}}\left[-4,4\right]\text{\hspace{0.17em}}$ for the following values of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b.$

1. $a=2;b=3$
2. $a=2;b=4$
3. $a=2;b=–4$
4. $a=2;b=–5$

## Extensions

Find the value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ if a linear function goes through the following points and has the following slope: $\text{\hspace{0.17em}}\left(x,2\right),\left(-4,6\right),\text{\hspace{0.17em}}m=3$

Find the value of y if a linear function goes through the following points and has the following slope: $\text{\hspace{0.17em}}\left(10,y\right),\left(25,100\right),\text{\hspace{0.17em}}m=-5$

y = 175

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