Lesson 6: Transformations of Graphs of Linear Functions

Property

The graph of $f(x) = x + k$ shifts the graph of $f(x) = x$ vertically $k$ units.

• If $k > 0$, shift the line vertically up $k$ units.
• If $k < 0$, shift the line vertically down $|k|$ units.

Examples

Property

The graph of $g(x) = f(x + h)$ shifts the graph of $f(x) = x$ horizontally $h$ units.

• If $h > 0$, shift the line horizontally left $h$ units.
• If $h < 0$, shift the line horizontally right $|h|$ units.

Examples

Property

For the parent function $f(x) = x$, horizontal stretches and shrinks are created using $f(ax)$ where $a$ is a positive constant. When $a > 1$, the graph is horizontally compressed (shrunk) by a factor of $\frac{1}{a}$. When $0 < a < 1$, the graph is horizontally stretched by a factor of $\frac{1}{a}$.

Examples

Property

The coefficient $a$ in the function $f(x) = ax$ affects the graph of $f(x) = x$ by stretching or compressing it.

• If $0 < |a| < 1$, the graph of $f(x) = ax$ will be less steep than the graph of $f(x) = x$.
• If $|a| > 1$, the graph of $f(x) = ax$ will be steeper than the graph of $f(x) = x$.

Examples