2.6: Transformations of Functions

This section describes how to transform parent functions, which are simple functions you will encounter.

Parent Functions

Parent functions are functions that have different shapes, based on their formula. Memorize the following.

$\large f(x) = x$ $\large f(x) = x^2$ $\large f(x) = \sqrt{x}$ $\large f(x) = \lvert x \rvert$

$\large f(x) = x$ $\large f(x) = x^2$

$\large f(x) = \sqrt{x}$ $\large f(x) = \lvert x \rvert$

In transformations, every line in the plane can be transformed from $f(x) = x$. Let's see how.

Vertical Shift

Vertical Shifts of Graphs

To graph $y = f(x) + c$, shift $f(x)$ upwards $c$ units. To graph $y = f(x) - c$, shift $f(x)$ downwards $c$ units.

Vertical Shifts of Graphs

To graph $y = f(x) + c$, shift $f(x)$ upwards $c$ units.

To graph $y = f(x) - c$, shift $f(x)$ downwards $c$ units.

To identify transformations, remove every number except for the variable $x$. The result is the parent!

Identify the parent function and the transformation(s):

$g(x) = x^2 + 2$

Parent is $f(x) = x^2$.

Transformation is vertical shift up 2 units because $g(x) = f(x) + 2 = x^2 + 2$.
$g(x) = -2 + \sqrt{x}$

Parent is $f(x) = \sqrt{x}$.

Transformation is vertical shift down 2 units because \begin{align} g(x) &= f(x) -2 \\&= \sqrt{x} - 2 \\&= -2 + \sqrt{x} \end{align}

Horizontal Shift

Horizontal Shifts of Graphs

To graph $y = f(x - c)$, shift $f(x)$ right $c$ units. To graph $y = f(x + c)$, shift $f(x)$ left $c$ units.

Horizontal Shifts of Graphs

To graph $y = f(x - c)$, shift $f(x)$ right $c$ units.

To graph $y = f(x + c)$, shift $f(x)$ left $c$ units.

Identify the parent function and the transformation(s):

$g(x) = (x-3)^2$

Parent is $f(x) = x^2$.

Transformation is horizontal shift right 3 units because $g(x) = f(x-3)$.
$g(x) = -3 + \lvert x + 2\rvert$

Parent is $f(x) = \lvert x \rvert$.

Transformations are vertical shift down 3 units and horizontal shift left 2 units because $g(x) = -3 + f(x + 2)$.

Reflection

Reflecting Graphs

To graph $y = -f(x)$, reflect $f(x)$ around the $x$-axis. To graph $y = f(-x)$, reflect $f(x)$ around the $y$-axis.

Reflecting Graphs

To graph $y = -f(x)$, reflect $f(x)$ around the $x$-axis.

To graph $y = f(-x)$, reflect $f(x)$ around the $y$-axis.

If there is a horizontal shift + another horizontal transformation, make sure the coefficient of $x$ is $1$ before identifying transformations.

In particular, a reflection around the $y$-axis is a horizontal transformation.

Identify the parent function and the transformation(s):

$g(x) = -\sqrt{-x}$

Parent is $f(x) = \sqrt{x}$.

Transformations are reflection around $x$- and $y$-axis because $g(x) = -f(-x)$.
$g(x) = \sqrt{-x - 2}$

Parent is $f(x) = \sqrt{x}$.

The $-x$ seems like a reflection around the $y$-axis and the $-2$ a horizontal shift right 2 units. This is incorrect due to the presence of 2 horizontal transformations.

Factor out the $-$ to get $g(x) = \sqrt{-x - 2} = \sqrt{-(x+2)}$.

Transformations are reflection around the $y$-axis and horizontal shift left 2 units because $g(x) = f(-(x+2))$.

Vertical Stretch/Shrink

Vertical Stretching and Shrinking
To graph $y = c\cdot f(x)$:

If $c > 1$, stretch $f(x)$ vertically by a factor of $c$. If $0 < c < 1$, shrink $f(x)$ vertically by a factor of $c$.

Vertical Stretching and Shrinking
To graph $y = c\cdot f(x)$:

If $c > 1$, stretch $f(x)$ vertically by a factor of $c$.

If $0 < c < 1$, shrink $f(x)$ vertically by a factor of $c$.

Identify the parent function and the transformation(s):

$g(x) = \frac{1}{2}\sqrt{x - 2} + 2$
Parent is $f(x) = \sqrt{x}$.

Transformations are
- vertical shrink by a factor of $\frac{1}{2}$
- horizontal shift right 2 units
- vertical shift up 2 units
because $g(x) = \frac{1}{2}f(x - 2) + 2$.
$g(x) = 2(3 - x)^2$
Parent is $f(x) = x^2$.

$x$ is negative and there is also the term $3$, which are two horizontal transformations.

Factor out the $-$ to get \begin{align} g(x) &= 2(3-x)^2 \\&= 2(-x + 3)^2 \\&= 2\big[-(x - 3)\big]^2 \end{align}

Transformations are
- vertical stretch by a factor of 2
- $y$-axis reflection
- horizontal shift right 3 units
because $g(x) = 2f(-(x-3))$.

Horiztonal Stretch/Shrink

Horizontal Stretching and Shrinking
To graph $y = f(c\cdot x)$:

If $c > 1$, shrink $f(x)$ horizontally by a factor of $\frac{1}{c}$. If $0 < c < 1$, stretch $f(x)$ horizontally by a factor of $\frac{1}{c}$.

Horizontal Stretching and Shrinking
To graph $y = f(c\cdot x)$:

If $c > 1$, shrink $f(x)$ horizontally by a factor of $\frac{1}{c}$.

If $0 < c < 1$, stretch $f(x)$ horizontally by a factor of $\frac{1}{c}$.

Identify the parent function and the transformation(s):

$f(x) = -3 - \sqrt{\frac{1}{2}x}$
Parent is $f(x) = \sqrt{x}$.

Transformations are
- vertical shift down 3 units
- $x$-axis reflection
- horizontal stretch by a factor of $\dfrac{1}{\frac{1}{2}} = 2$ units
because $g(x) = -3-f\left(-\frac{1}{2}\right)$.
$f(x) = 1 - \left(3 - 6x\right)^2$
Parent is $f(x) = x^2$.

$x$ is negative, there is a term $3$ and a factor 6, which are three horizontal transformations.

Factor out the $-6$ to get \begin{align} g(x) &= 1 - \left(3 - 6x\right)^2 \\&= 1 - \left(-6x + 3\right)^2 \\&= 1 - \left[-6\left(x - \frac{1}{2}\right)\right]^2 \end{align}

Transformations are
- vertical shift up 1
- $x$-axis reflection
- $y$-axis reflection
- horizontal shrink by a factor of $\frac{1}{6}$
- horizontal shift right $\frac{1}{2}$ units
because $g(x) = 1 - f\left(-6\left(x - \frac{1}{2}\right)\right)$.

Graphing

Now that we know how to recognize transformations, graphing is straightforward.

To graph a transformed function, identify the parent, the transformations, and transform step by step!

Graph the function $f(x) = 1 - (x-2)^2$.

Graph the function $f(x) = -\sqrt{-2x - 2}$.

Graph the function $f(x) = -2\lvert{-x - 1}\rvert + 3$.