# Transformations of Graphs

### Parent Functions

The most basic function from a family of functions is called a parent function. Related functions can be graphed by modifying the graph of the parent function.

The method of graphing functions by plotting points is effective for a computer or calculator, which can evaluate functions quickly for a lot of points. However, it is not very efficient as a way to graph functions by hand.

A more efficient way is to understand some of the common types of functions and how changes to the rule of the function affect the graph. A parent function is a function from a certain category of functions with the simplest algebraic rule. Some common parent functions include the linear function $f(x)=x$, the quadratic function $f(x)=x^2$, and the cubic function $f(x)=x^3$.

When the algebraic rule of a parent function is manipulated in certain ways, there is a change, or transformation, of the graph of the function. Some examples of transformations are translations, stretches, compressions, and reflections.

### Translations of Graphs

The graph of a function can be translated vertically or horizontally by performing addition or subtraction within the function rule.

A translation, or shift, is a transformation in which a graph is moved vertically or horizontally. Adding or subtracting a constant to the input or output of a function will translate the graph in the coordinate plane without turning, flipping, or changing its shape.

When a constant is added to or subtracted from the output of the parent function, the graph is translated vertically.

Vertical Translations Example
For the graph of $f(x)$ and $k \gt 0$:
• The graph of $f(x)+k$ is a translation of $k$ units up.
• The graph of $f(x)-k$ is a translation of $k$ units down.
The example graph shows the parent function $f(x)$, its vertical translation up by 2 units as $f(x)+2$, and its vertical translation down by 2 units as $f(x)-2$.

When a constant is added to or subtracted from the input of the parent function, the graph is translated horizontally.
Horizontal Translations Example
For the graph of $f(x)$ and $h \gt 0$:
• The graph of $f(x-h)$ is a translation $h$ units to the right.
• The graph of $f(x+h)$ is a translation of $h$ units to the left.
Note that horizontal translations are opposite in direction from the sign. Subtracting $h$ from the input translates the graph in the positive direction; subtracting $h$ translates it in the negative direction.

The graph shows the parent function $f(x)$, its horizontal translation to the right as $f(x-2)$, and its horizontal translation to the left as $f(x+2)$.

Horizontal and vertical translations can also be combined. For example, the graph of $f(x-h)+k$ is a translation of $h$ units to the right and $k$ units up.

### Vertical Stretches and Compressions

The graph of a function can be stretched or compressed vertically or horizontally by performing multiplication within the function rule by a positive constant.

A stretch is a transformation that moves each point of a graph farther away from the $x$- or $y$-axis by a scale factor that is greater than 1.

A compression is a transformation that moves each point of a graph closer to the $x$- or $y$-axis by a scale factor that is between zero and 1.

When the output of the parent function is multiplied by a positive constant, the graph stretches or compresses vertically.

### Graphing Vertical Stretches and Compressions

Vertical Stretches and Compressions Example
For the graph of $f(x)$ and $a \gt 0$,
• The graph of $af(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of $a$ if $a \gt 1$.
• The graph of $af(x)$ is a vertical compression of the graph of $f(x)$ by a factor of $a$ if $0 \lt a \lt 1$.
The graph shows the parent function $f(x)$, its vertical stretch by a factor of 2 as $2f(x)$, and its vertical compression by $\frac{1}{4}$ as $\frac{1}{4}f(x)$.

### Reflections of Graphs

The graph of a function can be reflected across the $x$- or $y$-axis by performing multiplication within the function rule by –1.
A reflection is a transformation in which a graph is flipped across a line. When the output of the parent function is multiplied by –1, the graph is reflected across the x-axis. Multiplying by –1 changes all of the positive $y$-values to negative values. It also changes all of the negative $y$-values to positive values. The result is a graph that looks like it is "flipped" across the $x$-axis.
Reflections Across the $x$-Axis Example
For the graph of $f(x)$, the graph of $-f(x)$ is a reflection of the graph of $f(x)$ across the $x$-axis.

The graph shows the relation of the parent function $f(x)=x^2$ and its reflection across the $x$-axis as $f(x)=-x^2$.

### Combined Transformations

The graph of a function can be transformed by using a combination of translations, stretches, compressions, and reflections.

When multiple transformations are applied to a function, it is important to perform the transformations in the correct order. Think of the order of operations: If operations are not performed in the correct order, the results may be different. Also, like the order of operations, transformations involving multiplication should be performed before addition and subtraction.

Order of Transformations
1. Identify the parent function.
2. Perform any reflections.
3. Perform any stretches or compressions.
4. Perform any translations.

Step-By-Step Example
Graphing a Function with Combined Transformations
Graph the function:
$f(x)=3(x-2)^2+8$
Step 1

Identify the parent function.

The given function is:
$f(x)=3(x-2)^{\color{#c42126}2}+8$
The parent function of the given function is $y=x^2$.
Step 2

Perform any reflections.

The parent function is multiplied by 3, which is positive.
$f(x)={\color{#c42126}3}(x-2)^2+8$
So, the graph of the given function does not have any reflections.
Step 3

Perform any stretches or compressions.

The parent function is multiplied by 3, so the graph of the function is stretched vertically by a factor of 3:
$f(x)={\color{#c42126}3}(x-2)^2+8$
Graph the function $f(x)=3x^2$ to show the stretch. For stretches and compressions, it is helpful to look at a point such as $(1,1)$ on the parent function graph. After the stretch, the point is at $(1,3)$.
Solution

Perform any translations.

The given function contains a subtraction of 2 and an addition of 8.
$f(x)=3(x{\color{#c42126}{\;-\;2}})^2{\color{#0047af}{\;+\;8}}$
The –2 represents a translation of 2 units to the right. The positive 8 represents a translation of 8 units up. So, point $(0,0)$ is translated to point $(2,8)$.