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## Subsection1.9Function Transformations

It is also useful to talk about transformations of functions. Several key facts will be useful.

###### Function Transformations
1. Multiplying a function by a constant, $c\text{,}$ stretches the graph vertically (if $c \gt 1$) or shrinks the graph vertically (if $0 \lt c \lt 1$)
2. A negative sign (if $c \lt 0$) reflects the graph about the $x$-axis, in addition to shrinking or stretching.
3. Replacing $y$ by $(y+k)$ moves a graph up by $k$ (down if $k$ is negative).
4. Replacing $x$ by $(x-h)$ moves a graph to the right by $h$ (to the left if $h$ is negative).
###### Example1.20

Let $f(x)=x^2\text{.}$ We will explore different transformations performed on the graph of $f(x)\text{.}$

1. $g(x)=2\cdot f(x)=2x^2$ is a vertical stretch of $f(x)$ by a factor of 2.

2. $h(x)=-2\cdot f(x)=-2x^2$ is a vertical stretch of $f(x)$ by a factor of 2 and a reflection across the $x$-axis.

3. $j(x)=f(x)+5=x^2+5$ is a vertical shift of $f(x)$ up 5 units.

4. $k(x)=f(x+5)=(x+5)^2$ is a horizontal shift of $f(x)$ left 5 units.

###### Order of Transformations

Suppose that $f(x)$ is our function we are applying transformations to. Once written in the form

\begin{equation*} a\cdot f(b\cdot (x+h))+k \end{equation*}

the order of transformations is:

1. horizontal stretch or compress by a factor of $|b|$ or $|\dfrac{1}{b}|$ (if $b\lt 0$ then also reflect about $y$-axis)
2. shift horizontally left/ right by $|h|$
3. vertically stretch or compress by a factor of $|a|$ or $|\dfrac{1}{a}|$ (if $a\lt 0$ then also reflect about $x$-axis)
4. shift vertically up/ down by $|k|$
###### Example1.21

Describe the function

\begin{equation*} 5\cdot [f(-3x-6)-1] \end{equation*}

as a list of transformations done to $f(x)$ in the appropriate order.

Solution

First, let's rewrite the function in the form given above: \begin{align*} 5\cdot [f(-3x-6)-1]\amp = 5f(-3x-6)-5 \ \ \text{we distributed in the 5}\\ \amp = 5f(-3(x+2))-5 \ \ \text{we factored out the -3} \end{align*} Now that the function is written in the desired order, we may list off the transformations in the correct order using the order discussed above:

1. horizontal compress by $3$ and reflect about the $y$-axis
2. shift horizontally left by $2$
3. vertically stretch by $5$
4. shift vertically down by $5$