Absolute Value Graph (y = a|x-h| + k) – Tutorial

Absolute Value Graph: y = a|x-h| + k (In-Depth Explanation)

1) What does absolute value mean?

The expression |x-h| represents the distance from x to the point h. It is always 0 or positive.

2) Why is the vertex at (h, k)?

When x = h, |x-h| = 0. Therefore:

y(h) = a·0 + k = k

So the vertex is at (h, k). The "corner" of the absolute value graph occurs because |x-h| changes direction at x = h.

3) Axis of symmetry

|x-h| is symmetric about h. Points equally distant from h on either side give the same y value:

|(h-d)-h| = |(h+d)-h| = d

So the axis of symmetry of the graph is the line x = h.

4) What does the coefficient a do?

• a > 0: Graph opens upward (V shape).
• a < 0: Graph opens downward (∧ shape).
• Larger |a|: Arms become steeper (graph "narrows").
• Smaller |a|: Arms become flatter (graph "widens").

5) Piecewise representation

The absolute value expands as:

• If x ≥ h: |x-h| = x-h
• If x < h: |x-h| = -(x-h) = h-x

Therefore:

y = a|x-h| + k =

{ a(x-h)+k,   x≥h
a(h-x)+k,   x<h }

6) Slope interpretation

In the region x ≥ h, the slope is +a; in the region x < h, the slope is -a. This is why there is a "break/corner" in the graph at the vertex.

7) Intercepts

• y-intercept: Substitute x=0 (if 0 is within the range).
• x-intercept: Solve a|x-h|+k=0 for y=0:
  • If a=0: y=k is a constant line
  • If a≠0: |x-h| = -k/a. If the right side is negative, there are no roots.

Note: This tool plots the graph using numerical sampling, but the vertex and symmetry information are derived analytically from the function structure.