# 4.1 Linear functions  (Page 10/27)

 Page 10 / 27
$y=2x+4$

Given a graph of linear function, find the equation to describe the function.

1. Identify the y- intercept of an equation.
2. Choose two points to determine the slope.
3. Substitute the y- intercept and slope into the slope-intercept form of a line.

## Matching linear functions to their graphs

Match each equation of the linear functions with one of the lines in [link] .

Analyze the information for each function.

1. This function has a slope of 2 and a y -intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ has the same slope, but a different y- intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through $\text{\hspace{0.17em}}\left(0,\text{3}\right)\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ must be represented by line I.
2. This function also has a slope of 2, but a y -intercept of $\text{\hspace{0.17em}}-3.\text{\hspace{0.17em}}$ It must pass through the point $\text{\hspace{0.17em}}\left(0,-3\right)\text{\hspace{0.17em}}$ and slant upward from left to right. It must be represented by line III.
3. This function has a slope of –2 and a y- intercept of 3. This is the only function listed with a negative slope, so it must be represented by line IV because it slants downward from left to right.
4. This function has a slope of $\text{\hspace{0.17em}}\frac{1}{2}\text{\hspace{0.17em}}$ and a y- intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. Lines I and II pass through $\text{\hspace{0.17em}}\left(0,\text{3}\right),$ but the slope of $\text{\hspace{0.17em}}j\text{\hspace{0.17em}}$ is less than the slope of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ so the line for $\text{\hspace{0.17em}}j\text{\hspace{0.17em}}$ must be flatter. This function is represented by Line II.

Now we can re-label the lines as in [link] .

## Finding the x -intercept of a line

So far we have been finding the y- intercepts of a function: the point at which the graph of the function crosses the y -axis. Recall that a function may also have an x -intercept , which is the x -coordinate of the point where the graph of the function crosses the x -axis. In other words, it is the input value when the output value is zero.

To find the x -intercept, set a function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ equal to zero and solve for the value of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ For example, consider the function shown.

$f\left(x\right)=3x-6$

Set the function equal to 0 and solve for $\text{\hspace{0.17em}}x.$

$\begin{array}{ccc}\hfill 0& =& 3x-6\hfill \\ \hfill 6& =& 3x\hfill \\ \hfill 2& =& x\hfill \\ \hfill x& =& 2\hfill \end{array}$

The graph of the function crosses the x -axis at the point $\text{\hspace{0.17em}}\left(2,\text{0}\right).$

Do all linear functions have x -intercepts?

No. However, linear functions of the form $\text{\hspace{0.17em}}y=c,$ where $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is a nonzero real number are the only examples of linear functions with no x-intercept. For example, $\text{\hspace{0.17em}}y=5\text{\hspace{0.17em}}$ is a horizontal line 5 units above the x-axis. This function has no x-intercepts, as shown in [link] .

## x -intercept

The x -intercept of the function is value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}f\left(x\right)=0.\text{\hspace{0.17em}}$ It can be solved by the equation $\text{\hspace{0.17em}}0=mx+b.$

## Finding an x -intercept

Find the x -intercept of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{2}x-3.$

Set the function equal to zero to solve for $\text{\hspace{0.17em}}x.$

$\begin{array}{ccc}\hfill 0& =& \frac{1}{2}x-3\hfill \\ \hfill 3& =& \frac{1}{2}x\hfill \\ \hfill 6& =& x\hfill \\ \hfill x& =& 6\hfill \end{array}$

The graph crosses the x -axis at the point $\text{\hspace{0.17em}}\left(6,\text{0}\right).$

Find the x -intercept of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{4}x-4.$

## Describing horizontal and vertical lines

There are two special cases of lines on a graph—horizontal and vertical lines. A horizontal line    indicates a constant output, or y -value. In [link] , we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use $\text{\hspace{0.17em}}m=0\text{\hspace{0.17em}}$ in the equation $\text{\hspace{0.17em}}f\left(x\right)=mx+b,$ the equation simplifies to $\text{\hspace{0.17em}}f\left(x\right)=b.\text{\hspace{0.17em}}$ In other words, the value of the function is a constant. This graph represents the function $\text{\hspace{0.17em}}f\left(x\right)=2.$

find general solution of the Tanx=-1/root3,secx=2/root3
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
where can I get indices
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Need help with this question please
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
yah
immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
how would this look as an equation?
Hayden
5x+x=45
Khay
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n