# Transforming Linear Graphs

### Translations of Linear Functions

A translation of a linear function shifts its graph vertically or horizontally.

The parent function, or most basic function in the linear family, is the linear function $f(x) = x$. Its graph is a line that passes through the origin and has a slope of $m=1$. Other linear functions can be graphed as a transformation of the parent function $f(x) = x$.

A translation, or shift, is a transformation in which a graph is moved vertically or horizontally. For any function $f(x)$, the function can be translated vertically $k$ units or horizontally $h$ units.

For $h\gt0$ and $k\gt0$:

• The graph of $f(x)+k$ is the graph of $f(x)$ translated up $k$ units.
• The graph of $f(x)-k$ is the graph of $f(x)$ translated down $k$ units.
• The graph of $f(x-h)$ is the graph of $f(x)$ translated right $h$ units.
• The graph of $f(x+h)$ is the graph of $f(x)$ translated left $h$ units.

A translation of the parent linear function $f(x) = x$ can be viewed as either a horizontal or vertical translation.

For example, the parent function of a line is:
$f(x) = x$
Its $y$-intercept is zero. Its $x$-intercept is also zero. One translation of the parent function is:
$f(x)=x+4$
It is a vertical translation of 4 units up or a horizontal translation 4 units to the left. The $y$-intercept is 4. The $x$-intercept is –4. Another translation is:
$f(x)=x-5$
It is a vertical translation 5 units down or a horizontal translation 5 units to the right. The $y$-intercept is –5. The $x$-intercept is 5. All functions have a slope of 1.

### Stretches, Compressions, and Reflections of Linear Functions

A stretch or compression of the graph of a linear function is a pull away from an axis or a push toward an axis. A reflection of the graph of a linear function is a flip across a line.

For any function $f(x)$, the function can be stretched, compressed, or reflected across an axis. A reflection is a transformation in which a figure is flipped across a line.

For $a>0$, where $a$ is a scaling factor:

• The graph of $af(x)$ is a vertical stretch or compression of the graph of $f(x)$ by a factor of $a$. For $a \gt 1$, the graph of $f(x)$ is stretched. For $0\lt a\lt 1$, the graph of $f(x)$ is compressed.
• The graph of $-f(x)$ is the reflection of the graph of $f(x)$ across the $x$-axis.
For example, the parent function of a line is:
$f(x) = x$
Its slope is 1. The function that shows the vertical stretch of the parent function by a factor of 4 is:
$f(x)=4x$
The slope of the vertical stretch function is 4. The function that shows a vertical compression of the parent function by a factor of $\frac{1}{4}$ is:
$f(x)=\frac{1}{4}x$
The slope of the vertical compression function is $\frac{1}{4}$.
The function that shows a reflection across the $x$-axis is:
$f(x)=-x$
The slope of the reflection function is –1. All three functions have a $y$-intercept of zero.

### Combined Transformations of Linear Functions

Some linear functions are the result of multiple transformations of the linear parent function.
Sometimes, a parent function may be transformed in more than one way. In general, the order of transformations is like the order of operations: Perform stretches, compressions, and reflections first (multiplication and division), followed by translations (addition and subtraction).
Step-By-Step Example
Graphing a Linear Function with Combined Transformations
Graph the given linear function by using transformations of its parent function $f(x)=x$:
$f(x)=2x+3$
Step 1
Identify the types of transformations of the parent function $f(x)=x$ that are used in the given function:
$f(x)=2x+3$
The first term in the linear function is $2x$. It shows that the parent function is multiplied by 2, so the graph will be stretched vertically by a factor of 2.

The next term in the given linear function is 3. It shows that 2 is added to the vertically stretched function f(x)=2x. So, the vertically stretched graph will be translated 3 units up.

Step 2
Graph the first transformation:
$f(x)=2x$
The graph of the parent function is vertically stretched by a factor of 2.
Solution
Then graph the second transformation. The function $f(x)=2x$ is translated up 3 units. This is represented by the graph of the function:
$f(x)=2x+3$