# 3.5 Transformation of functions  (Page 11/21)

 Page 11 / 21

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x -axis from a reflection with respect to the y -axis?

How can you determine whether a function is odd or even from the formula of the function?

For a function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ substitute $\text{\hspace{0.17em}}\left(-x\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}\left(x\right)\text{\hspace{0.17em}}$ in $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ Simplify. If the resulting function is the same as the original function, $\text{\hspace{0.17em}}f\left(-x\right)=f\left(x\right),\text{\hspace{0.17em}}$ then the function is even. If the resulting function is the opposite of the original function, $\text{\hspace{0.17em}}f\left(-x\right)=-f\left(x\right),\text{\hspace{0.17em}}$ then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

## Algebraic

For the following exercises, write a formula for the function obtained when the graph is shifted as described.

$\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ is shifted up 1 unit and to the left 2 units.

$\text{\hspace{0.17em}}f\left(x\right)=|x|\text{\hspace{0.17em}}$ is shifted down 3 units and to the right 1 unit.

$g\left(x\right)=|x-1|-3$

$\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ is shifted down 4 units and to the right 3 units.

$\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{x}^{2}}\text{\hspace{0.17em}}$ is shifted up 2 units and to the left 4 units.

$g\left(x\right)=\frac{1}{{\left(x+4\right)}^{2}}+2$

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function $\text{\hspace{0.17em}}f.$

$y=f\left(x-49\right)$

$y=f\left(x+43\right)$

The graph of $\text{\hspace{0.17em}}f\left(x+43\right)\text{\hspace{0.17em}}$ is a horizontal shift to the left 43 units of the graph of $\text{\hspace{0.17em}}f.$

$y=f\left(x+3\right)$

$y=f\left(x-4\right)$

The graph of $\text{\hspace{0.17em}}f\left(x-4\right)\text{\hspace{0.17em}}$ is a horizontal shift to the right 4 units of the graph of $\text{\hspace{0.17em}}f.$

$y=f\left(x\right)+5$

$y=f\left(x\right)+8$

The graph of $\text{\hspace{0.17em}}f\left(x\right)+8\text{\hspace{0.17em}}$ is a vertical shift up 8 units of the graph of $\text{\hspace{0.17em}}f.$

$y=f\left(x\right)-2$

$y=f\left(x\right)-7$

The graph of $\text{\hspace{0.17em}}f\left(x\right)-7\text{\hspace{0.17em}}$ is a vertical shift down 7 units of the graph of $\text{\hspace{0.17em}}f.$

$y=f\left(x-2\right)+3$

$y=f\left(x+4\right)-1$

The graph of $f\left(x+4\right)-1$ is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of $f.$

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

$f\left(x\right)=4{\left(x+1\right)}^{2}-5$

$g\left(x\right)=5{\left(x+3\right)}^{2}-2$

decreasing on $\text{\hspace{0.17em}}\left(-\infty ,-3\right)\text{\hspace{0.17em}}$ and increasing on $\text{\hspace{0.17em}}\left(-3,\infty \right)$

$a\left(x\right)=\sqrt{-x+4}$

$k\left(x\right)=-3\sqrt{x}-1$

decreasing on $\left(0,\text{\hspace{0.17em}}\infty \right)$

## Graphical

For the following exercises, use the graph of $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ shown in [link] to sketch a graph of each transformation of $\text{\hspace{0.17em}}f\left(x\right).$

$g\left(x\right)={2}^{x}+1$

$h\left(x\right)={2}^{x}-3$

$w\left(x\right)={2}^{x-1}$

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

$f\left(t\right)={\left(t+1\right)}^{2}-3$

$h\left(x\right)=|x-1|+4$

$k\left(x\right)={\left(x-2\right)}^{3}-1$

$m\left(t\right)=3+\sqrt{t+2}$

## Numeric

Tabular representations for the functions $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ are given below. Write $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)\text{\hspace{0.17em}}$ as transformations of $\text{\hspace{0.17em}}f\left(x\right).$

 $x$ −2 −1 0 1 2 $f\left(x\right)$ −2 −1 −3 1 2
 $x$ −1 0 1 2 3 $g\left(x\right)$ −2 −1 −3 1 2
 $x$ −2 −1 0 1 2 $h\left(x\right)$ −1 0 −2 2 3

$g\left(x\right)=f\left(x-1\right),\text{\hspace{0.17em}}h\left(x\right)=f\left(x\right)+1$

Tabular representations for the functions $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ are given below. Write $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)\text{\hspace{0.17em}}$ as transformations of $\text{\hspace{0.17em}}f\left(x\right).$

 $x$ −2 −1 0 1 2 $f\left(x\right)$ −1 −3 4 2 1
 $x$ −3 −2 −1 0 1 $g\left(x\right)$ −1 −3 4 2 1
 $x$ −2 −1 0 1 2 $h\left(x\right)$ −2 −4 3 1 0

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

$f\left(x\right)=|x-3|-2$

$f\left(x\right)=\sqrt{x+3}-1$

$f\left(x\right)={\left(x-2\right)}^{2}$

$f\left(x\right)=|x+3|-2$

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

$f\left(x\right)=-\sqrt{x}$

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

$f\left(x\right)=-{\left(x+1\right)}^{2}+2$

$f\left(x\right)=\sqrt{-x}+1$

For the following exercises, determine whether the function is odd, even, or neither.

$f\left(x\right)=3{x}^{4}$

even

$g\left(x\right)=\sqrt{x}$

$h\left(x\right)=\frac{1}{x}+3x$

odd

$f\left(x\right)={\left(x-2\right)}^{2}$

$g\left(x\right)=2{x}^{4}$

even

$h\left(x\right)=2x-{x}^{3}$

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function $\text{\hspace{0.17em}}f.$

$g\left(x\right)=-f\left(x\right)$

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a vertical reflection (across the $\text{\hspace{0.17em}}x$ -axis) of the graph of $\text{\hspace{0.17em}}f.$

$g\left(x\right)=f\left(-x\right)$

$g\left(x\right)=4f\left(x\right)$

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a vertical stretch by a factor of 4 of the graph of $\text{\hspace{0.17em}}f.$

$g\left(x\right)=6f\left(x\right)$

$g\left(x\right)=f\left(5x\right)$

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a horizontal compression by a factor of $\text{\hspace{0.17em}}\frac{1}{5}\text{\hspace{0.17em}}$ of the graph of $\text{\hspace{0.17em}}f.$

$g\left(x\right)=f\left(2x\right)$

$g\left(x\right)=f\left(\frac{1}{3}x\right)$

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a horizontal stretch by a factor of 3 of the graph of $\text{\hspace{0.17em}}f.$

$g\left(x\right)=f\left(\frac{1}{5}x\right)$

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

The graph of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is a horizontal reflection across the $\text{\hspace{0.17em}}y$ -axis and a vertical stretch by a factor of 3 of the graph of $\text{\hspace{0.17em}}f.$

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

For the following exercises, write a formula for the function $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ that results when the graph of a given toolkit function is transformed as described.

The graph of $\text{\hspace{0.17em}}f\left(x\right)=|x|\text{\hspace{0.17em}}$ is reflected over the $\text{\hspace{0.17em}}y$ - axis and horizontally compressed by a factor of $\text{\hspace{0.17em}}\frac{1}{4}$ .

$g\left(x\right)=|-4x|$

The graph of $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ is reflected over the $\text{\hspace{0.17em}}x$ -axis and horizontally stretched by a factor of 2.

The graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{x}^{2}}\text{\hspace{0.17em}}$ is vertically compressed by a factor of $\text{\hspace{0.17em}}\frac{1}{3},\text{\hspace{0.17em}}$ then shifted to the left 2 units and down 3 units.

$g\left(x\right)=\frac{1}{3{\left(x+2\right)}^{2}}-3$

The graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

The graph of $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ is vertically compressed by a factor of $\text{\hspace{0.17em}}\frac{1}{2},\text{\hspace{0.17em}}$ then shifted to the right 5 units and up 1 unit.

$g\left(x\right)=\frac{1}{2}{\left(x-5\right)}^{2}+1$

The graph of $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

$g\left(x\right)=4{\left(x+1\right)}^{2}-5$

The graph of the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

$g\left(x\right)=5{\left(x+3\right)}^{2}-2$

$h\left(x\right)=-2|x-4|+3$

The graph of $\text{\hspace{0.17em}}f\left(x\right)=|x|\text{\hspace{0.17em}}$ is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

$k\left(x\right)=-3\sqrt{x}-1$

$m\left(x\right)=\frac{1}{2}{x}^{3}$

The graph of the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ is compressed vertically by a factor of $\text{\hspace{0.17em}}\frac{1}{2}.$

$n\left(x\right)=\frac{1}{3}|x-2|$

$p\left(x\right)={\left(\frac{1}{3}x\right)}^{3}-3$

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

$q\left(x\right)={\left(\frac{1}{4}x\right)}^{3}+1$

$a\left(x\right)=\sqrt{-x+4}$

The graph of $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ is shifted right 4 units and then reflected across the vertical line $\text{\hspace{0.17em}}x=4.$

For the following exercises, use the graph in [link] to sketch the given transformations.

$g\left(x\right)=f\left(x\right)-2$

$g\left(x\right)=-f\left(x\right)$

$g\left(x\right)=f\left(x+1\right)$

$g\left(x\right)=f\left(x-2\right)$

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar