# 3.5 Transformation of functions

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In this section, you will:
• Graph functions using vertical and horizontal shifts.
• Graph functions using reflections about the $\text{\hspace{0.17em}}x\text{-axis}\text{\hspace{0.17em}}$ axis and the $\text{\hspace{0.17em}}y\text{-axis}.$
• Determine whether a function is even, odd, or neither from its graph.
• Graph functions using compressions and stretches.
• Combine transformations.

We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.

## Graphing functions using vertical and horizontal shifts

Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.

## Identifying vertical shifts

One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift , moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function $\text{\hspace{0.17em}}g\left(x\right)=f\left(x\right)+k,\text{\hspace{0.17em}}$ the function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is shifted vertically $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ units. See [link] for an example.

To help you visualize the concept of a vertical shift, consider that $\text{\hspace{0.17em}}y=f\left(x\right).\text{\hspace{0.17em}}$ Therefore, $\text{\hspace{0.17em}}f\left(x\right)+k\text{\hspace{0.17em}}$ is equivalent to $\text{\hspace{0.17em}}y+k.\text{\hspace{0.17em}}$ Every unit of $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is replaced by $\text{\hspace{0.17em}}y+k,\text{\hspace{0.17em}}$ so the y -value increases or decreases depending on the value of $\text{\hspace{0.17em}}k.\text{\hspace{0.17em}}$ The result is a shift upward or downward.

## Vertical shift

Given a function $f\left(x\right),$ a new function $g\left(x\right)=f\left(x\right)+k,$ where $\text{\hspace{0.17em}}k$ is a constant, is a vertical shift    of the function $f\left(x\right).$ All the output values change by $k$ units. If $k$ is positive, the graph will shift up. If $k$ is negative, the graph will shift down.

## Adding a constant to a function

To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. [link] shows the area of open vents $\text{\hspace{0.17em}}V\text{\hspace{0.17em}}$ (in square feet) throughout the day in hours after midnight, $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.

We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in [link] .

Notice that in [link] , for each input value, the output value has increased by 20, so if we call the new function $\text{\hspace{0.17em}}S\left(t\right),$ we could write

$S\left(t\right)=V\left(t\right)+20$

This notation tells us that, for any value of $\text{\hspace{0.17em}}t,S\left(t\right)\text{\hspace{0.17em}}$ can be found by evaluating the function $\text{\hspace{0.17em}}V\text{\hspace{0.17em}}$ at the same input and then adding 20 to the result. This defines $\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ as a transformation of the function $\text{\hspace{0.17em}}V,\text{\hspace{0.17em}}$ in this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change. See [link] .

 $t$ 0 8 10 17 19 24 $V\left(t\right)$ 0 0 220 220 0 0 $S\left(t\right)$ 20 20 240 240 20 20

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
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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
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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