AP Calculus BC

Transforming Functions and Graphing Piecewise Functions

Piecewise functions

A piecewise function uses a different rule on different parts of its domain. To evaluate it at a point, find which piece's domain restriction contains that point, then use that piece's rule.

$C(t) = \begin{cases} 7 & 0 \le t \le 3 \\ 7 + t & 3 < t \le 8 \\ 18 & 8 < t \le 28 \end{cases}$

Transformations

Every function built from a parent function $p(x)$ as $a \cdot p(x - h) + k$ is that parent function transformed:

• $h$ shifts the graph horizontally: right if $h > 0$, left if $h < 0$.
• $k$ shifts the graph vertically: up if $k > 0$, down if $k < 0$.
• $a$ stretches or compresses the graph vertically by a factor of $|a|$, and reflects it over the x-axis if $a$ is negative.
• Replacing $x$ with $-x$ reflects the graph over the y-axis instead.

A transformation moves individual points the same way it moves the whole graph: a horizontal shift changes only the x-coordinate, a vertical shift or a reflection over the x-axis changes only the y-coordinate, and a reflection over the y-axis negates the x-coordinate.