### Translations of Linear Functions

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

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.

The function that shows a reflection across the $x$-axis is:

### Combined Transformations of Linear Functions

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.