Absolute Value Function Transformations

Transformations of the absolute value function

The parent function $y = |x|$ makes a V-shape with vertex at the origin. The general form $y = a|x - h| + k$ transforms it:

• $h$: shifts the vertex horizontally (right if positive, left if negative). Note: $|x - 3|$ shifts RIGHT to $x = 3$.
• $k$: shifts the vertex vertically (up if positive, down if negative).
• $|a|$: stretches ($|a| > 1$ makes it narrower) or compresses ($|a| < 1$ makes it wider).
• sign of $a$: negative $a$ flips the V upside-down (opens downward).

Step-by-step: Graph $y = -2|x + 1| + 3$

Step 1 — Vertex: $h = -1$, $k = 3$, so vertex is at $(-1, 3)$.

Step 2 — Direction: $a = -2 < 0$, so V opens downward.

Step 3 — Steepness: $|a| = 2$, so the arms are steeper than the basic $|x|$.

Step 4 — Plot: Start at vertex $(-1, 3)$. Going right 1 unit, $y$ decreases by 2 → point $(0, 1)$. Going left 1 unit, same → point $(-2, 1)$.

Try it in the visualization

Drag the $h$, $k$, $a$ sliders to see the V-shape shift, stretch, and flip in real time. The base function $y = |x|$ is shown as a dashed reference. The vertex is vertically, $|a|$ stretches, and $\text{sign}(a)$ flips. The vertex moves to $(h, k)$.