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

 Page 10 / 15

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

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?

what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
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?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?