Piecewise and Absolute Value Functions

What is a Piecewise Function?

A piecewise function is a function made up of different rules (or "pieces") applied to different parts of its domain. Instead of a single equation, the function looks at the input value ($x$) to decide which rule to use.

Example: Evaluate the following function at $x = -3$ and $x = 2$. $f(x) = \begin{cases} 2x & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$

• For $x = -3$: Since $-3 < 0$, we fall into the first interval. We use the top rule: $2x$. $f(-3) = 2(-3) = -6$
• For $x = 2$: Since $2 \geq 0$, we fall into the second interval. We use the bottom rule: $x^2$. $f(2) = (2)^2 = 4$

The Absolute Value Function

The absolute value function, written as $y = |x|$, represents the distance of a number from zero on the number line. Because distance is always positive (or zero), the graph of $y = |x|$ forms a sharp "V" shape with its vertex at the origin $(0,0)$.

Interestingly, the absolute value function is just a specific piecewise function: $f(x) = \begin{cases} -x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$

Transforming Absolute Value Functions

You can shift, stretch, or flip the V-shaped graph using the general transformation formula: $y = a|x - h| + k$

• $h$ (Horizontal Shift): Moves the graph left or right. The vertex's x-coordinate is $h$. (Note the minus sign: $|x - 2|$ moves right, $|x + 2|$ moves left).
• $k$ (Vertical Shift): Moves the graph up or down. The vertex's y-coordinate is $k$.
• $a$ (Stretch/Compression): Makes the "V" narrower or wider. If $a$ is negative, the "V" flips upside down.

Example: Graph $f(x) = |x - 2| + 1$.

1. Find the vertex: Based on $y = a|x - h| + k$, we have $h = 2$ and $k = 1$. The vertex is at $(2, 1)$.
2. Determine the shape: Since $a = 1$ (positive), the graph opens upwards with a standard slope of $1$ on the right side of the vertex and $-1$ on the left.
3. Plot points: Starting from the vertex $(2,1)$, go right 1 and up 1 to the point $(3,2)$. Go left 1 and up 1 to the point $(1,2)$. Connect them to form the V-shape.