# 4.1 Linear functions  (Page 9/27)

 Page 9 / 27

## Vertical stretch or compression

In the equation $\text{\hspace{0.17em}}f\left(x\right)=mx,$ the $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ is acting as the vertical stretch    or compression of the identity function. When $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ is negative, there is also a vertical reflection of the graph. Notice in [link] that multiplying the equation of $\text{\hspace{0.17em}}f\left(x\right)=x\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ stretches the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ by a factor of $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}m>\text{1}\text{\hspace{0.17em}}$ and compresses the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ by a factor of $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}0 This means the larger the absolute value of $\text{\hspace{0.17em}}m,\text{\hspace{0.17em}}$ the steeper the slope.

## Vertical shift

In $\text{\hspace{0.17em}}f\left(x\right)=mx+b,$ the $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ acts as the vertical shift    , moving the graph up and down without affecting the slope of the line. Notice in [link] that adding a value of $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ to the equation of $\text{\hspace{0.17em}}f\left(x\right)=x\text{\hspace{0.17em}}$ shifts the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ a total of $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ units up if $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is positive and $\text{\hspace{0.17em}}|b|\text{\hspace{0.17em}}$ units down if $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is negative.

Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.

Given the equation of a linear function, use transformations to graph the linear function in the form $\text{\hspace{0.17em}}f\left(x\right)=mx+b.$

1. Graph $\text{\hspace{0.17em}}f\left(x\right)=x.$
2. Vertically stretch or compress the graph by a factor $\text{\hspace{0.17em}}m.$
3. Shift the graph up or down $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ units.

## Graphing by using transformations

Graph $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{2}x-3\text{\hspace{0.17em}}$ using transformations.

The equation for the function shows that $\text{\hspace{0.17em}}m=\frac{1}{2}\text{\hspace{0.17em}}$ so the identity function is vertically compressed by $\text{\hspace{0.17em}}\frac{1}{2}.\text{\hspace{0.17em}}$ The equation for the function also shows that $\text{\hspace{0.17em}}b=-3\text{\hspace{0.17em}}$ so the identity function is vertically shifted down 3 units. First, graph the identity function, and show the vertical compression as in [link] .

Then show the vertical shift as in [link] .

Graph $\text{\hspace{0.17em}}f\left(x\right)=4+2x\text{\hspace{0.17em}}$ using transformations.

In [link] , could we have sketched the graph by reversing the order of the transformations?

No. The order of the transformations follows the order of operations. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. This is why we performed the compression first. For example, following the order: Let the input be 2.

$\begin{array}{ccc}\hfill f\left(2\right)& =& \frac{1}{2}\left(2\right)-3\hfill \\ & =& 1-3\hfill \\ & =& -2\hfill \end{array}$

## Writing the equation for a function from the graph of a line

Earlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at [link] . We can see right away that the graph crosses the y -axis at the point $\text{\hspace{0.17em}}\left(0,\text{4}\right)\text{\hspace{0.17em}}$ so this is the y -intercept.

Then we can calculate the slope by finding the rise and run. We can choose any two points, but let’s look at the point $\text{\hspace{0.17em}}\left(–2,0\right).\text{\hspace{0.17em}}$ To get from this point to the y- intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be

$m=\frac{\text{rise}}{\text{run}}=\frac{4}{2}=2$

Substituting the slope and y- intercept into the slope-intercept form of a line gives

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function