# 1.5 Transformation of functions  (Page 11/22)

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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

Write a formula for the function obtained when the graph of $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ is shifted up 1 unit and to the left 2 units.

Write a formula for the function obtained when the graph of $\text{\hspace{0.17em}}f\left(x\right)=|x|\text{\hspace{0.17em}}$ is shifted down 3 units and to the right 1 unit.

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

Write a formula for the function obtained when the graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ is shifted down 4 units and to the right 3 units.

Write a formula for the function obtained when the graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{x}^{2}}\text{\hspace{0.17em}}$ 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

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

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
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with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
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This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
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what type of identity
Jeffrey
Confunction Identity
Barcenas
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meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim