# 4.1 Linear functions  (Page 16/27)

 Page 16 / 27

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

what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Wilson
what is algebra
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
I want to know partial fraction Decomposition.
classes of function in mathematics
divide y2_8y2+5y2/y2
wish i knew calculus to understand what's going on 🙂
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
24x^5
James
10x
Axmed
24X^5
Taieb
comment écrire les symboles de math par un clavier normal
SLIMANE