# 2.2 Graphs of linear functions  (Page 10/15)

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$\begin{array}{c}3y+x=12\\ -y=8x+1\end{array}$

neither parallel or perpendicular

$\begin{array}{c}3y+4x=12\\ -6y=8x+1\end{array}$

$\begin{array}{c}6x-9y=10\\ 3x+2y=1\end{array}$

perpendicular

$\begin{array}{c}y=\frac{2}{3}x+1\\ 3x+2y=1\end{array}$

$\begin{array}{c}y=\frac{3}{4}x+1\\ -3x+4y=1\end{array}$

parallel

For the following exercises, find the x - and y- intercepts of each equation

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

$g\left(x\right)=2x+4$

;

$h\left(x\right)=3x-5$

$k\left(x\right)=-5x+1$

;

$-2x+5y=20$

$7x+2y=56$

;

For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

• Line 1: Passes through $\left(0,6\right)$ and $\left(3,-24\right)$
• Line 2: Passes through $\left(-1,19\right)$ and $\left(8,-71\right)$
• Line 1: Passes through $\left(-8,-55\right)$ and $\left(10,\text{\hspace{0.17em}}89\right)$
• Line 2: Passes through $\left(9,-44\right)$ and $\left(4,-14\right)$

• Line 1: Passes through $\left(2,3\right)$ and $\left(4,-1\right)$
• Line 2: Passes through $\left(6,3\right)$ and $\left(8,5\right)$
• Line 1: Passes through $\left(1,7\right)$ and $\left(5,5\right)$
• Line 2: Passes through $\left(-1,-3\right)$ and $\left(1,1\right)$

• Line 1: Passes through $\left(0,5\right)$ and $\left(3,3\right)$
• Line 2: Passes through $\left(1,-5\right)$ and $\left(3,-2\right)$
• Line 1: Passes through $\left(2,5\right)$ and $\left(5,-1\right)$
• Line 2: Passes through $\left(-3,7\right)$ and $\left(3,-5\right)$

Write an equation for a line parallel to $f\left(x\right)=-5x-3$ and passing through the point

Write an equation for a line parallel to $g\left(x\right)=3x-1$ and passing through the point $\left(4,9\right).$

$g\left(x\right)=3x-3$

Write an equation for a line perpendicular to $h\left(t\right)=-2t+4$ and passing through the point

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

$p\left(t\right)=-\frac{1}{3}t+2$

Find the point at which the line $f\left(x\right)=-2x-1$ intersects the line $g\left(x\right)=-x.$

Find the point at which the line $f\left(x\right)=2x+5$ intersects the line $g\left(x\right)=-3x-5.$

$\left(-2,1\right)$

Use algebra to find the point at which the line intersects the line $h\left(x\right)=\frac{9}{4}x\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{73}{10}.$

Use algebra to find the point at which the line $f\left(x\right)=\frac{7}{4}x\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{457}{60}$ intersects the line $g\left(x\right)=\frac{4}{3}x\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{31}{5}.$

$\left(-\frac{17}{5},\frac{5}{3}\right)$

## Graphical

For the following exercises, 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 and y -intercept of

An x -intercept of and y -intercept of

A y -intercept of and slope $-\frac{3}{2}$

A y -intercept of and slope $\frac{2}{5}$

Passing through the points and

Passing through the points and

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

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

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

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

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

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

$p\left(t\right)=-2\text{\hspace{0.17em}}+\text{\hspace{0.17em}}3t$

$x=3$

$x=-2$

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

$q\left(x\right)=3$

$4x=-9y+36$

$\frac{x}{3}-\frac{y}{4}=1$

$3x-5y=15$

$3x=15$

$3y=12$

If $g\left(x\right)$ is the transformation of $f\left(x\right)=x$ after a vertical compression by $\frac{3}{4},$ a shift right by 2, and a shift down by 4

1. Write an equation for $g\left(x\right).$
2. What is the slope of this line?
3. Find the y- intercept of this line.

• $g\left(x\right)=0.75x-5.5\text{}$
• 0.75
• $\left(0,-5.5\right)$

If $g\left(x\right)$ is the transformation of $f\left(x\right)=x$ after a vertical compression by $\frac{1}{3},$ a shift left by 1, and a shift up by 3

1. Write an equation for $g\left(x\right).$
2. What is the slope of this line?
3. Find the y- intercept of this line.

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

$y=\text{3}$

$x=-3$

For the following exercises, find the point of intersection of each pair of lines if it exists. If it does not exist, indicate that there is no point of intersection.

$\begin{array}{c}y=\frac{3}{4}x+1\\ -3x+4y=12\end{array}$

no point of intersection

$\begin{array}{c}2x-3y=12\\ 5y+x=30\end{array}$

$\begin{array}{c}2x=y-3\\ y+4x=15\end{array}$

$\begin{array}{c}x-2y+2=3\\ x-y=3\end{array}$

$\begin{array}{c}5x+3y=-65\\ x-y=-5\end{array}$

## Extensions

Find the equation of the line parallel to the line $g\left(x\right)=-0.\text{01}x\text{\hspace{0.17em}}\text{+}\text{\hspace{0.17em}}\text{2}\text{.01}$ through the point

Find the equation of the line perpendicular to the line $g\left(x\right)=-0.\text{01}x\text{+2}\text{.01}$ through the point

$y=100x-98$

For the following exercises, use the functions

Find the point of intersection of the lines $f$ and $g.$

Where is $f\left(x\right)$ greater than $g\left(x\right)?$ Where is $g\left(x\right)$ greater than $f\left(x\right)?$

$x<\frac{1999}{201}x>\frac{1999}{201}$

## Real-world applications

A car rental company offers two plans for renting a car.

• Plan A: $30 per day and$0.18 per mile
• Plan B: $50 per day with free unlimited mileage How many miles would you need to drive for plan B to save you money? A cell phone company offers two plans for minutes. • Plan A:$20 per month and $1 for every one hundred texts. • Plan B:$50 per month with free unlimited texts.

How many texts would you need to send per month for plan B to save you money?

Less than 3000 texts

A cell phone company offers two plans for minutes.

• Plan A: $15 per month and$2 for every 300 texts.
• Plan B: $25 per month and$0.50 for every 100 texts.

How many texts would you need to send per month for plan B to save you money?

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