Transforming Linear Graphs

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: Its $y$-intercept is zero. Its $x$-intercept is also zero. One translation of the parent function is: 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: 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.

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: Its slope is 1. The function that shows the vertical stretch of the parent function by a factor of 4 is: 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: The slope of the vertical compression function is $\frac{1}{4}$.
The function that shows a reflection across the $x$-axis is: The slope of the reflection function is –1. All three functions have a $y$-intercept of zero.