##
Subsection1.9Function Transformations

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

###### Function Transformations

- Multiplying a function by a constant, \(c\text{,}\) stretches the graph vertically (if \(c \gt 1\)) or shrinks the graph vertically (if \(0 \lt c \lt 1\))
- A negative sign (if \(c \lt 0\)) reflects the graph about the \(x\)-axis, in addition to shrinking or stretching.
- Replacing \(y\) by \((y+k)\) moves a graph up by \(k\) (down if \(k\) is negative).
- 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 \(f(x)\text{.}\)

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

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

\(k(x)=f(x+5)=(x+5)^2\) 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:

- horizontal stretch or compress by a factor of \(|b|\) or \(|\dfrac{1}{b}|\) (if \(b\lt 0\) then also reflect about \(y\)-axis)
- shift horizontally left/ right by \(|h| \)
- vertically stretch or compress by a factor of \(|a|\) or \(|\dfrac{1}{a}|\) (if \(a\lt 0\) then also reflect about \(x\)-axis)
- 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.

SolutionFirst, 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:

- horizontal compress by \(3\) and reflect about the \(y\)-axis
- shift horizontally left by \(2 \)
- vertically stretch by \(5\)
- shift vertically down by \(5 \)