Graphing the General y = a|x − h| + k
Simple Explanation
The general absolute value function y = a|x − h| + k combines every transformation at once: a sets direction and steepness, while (h, k) sets the vertex location. Reading all three coefficients off the equation lets you sketch any absolute value graph directly.
Why Do We Need It?
This combines sections 6.1-6.3 into the one complete method you will use for every absolute value graph from here on, exactly the way section 5.4 combined the quadratic transformations.
See It
A narrow, upward-opening V-shaped graph with its vertex at (-1, -3)
Formula
Vertex Form of an Absolute Value Function
y = a|x − h| + k, vertex = (h, k)
Every absolute value function graph is a V-shape (or upside-down V) whose sharp corner — the vertex — sits at exactly (h, k), directly readable from the equation.
- a
- — controls the direction (sign) and narrowness (size) of the V, same role as in a parabola
- h
- — the horizontal shift — the x-coordinate of the vertex
- k
- — the vertical shift — the y-coordinate of the vertex
When to use it: Whenever you need to find the vertex, sketch the graph, or identify the transformations of an absolute value function directly from its equation.
Worked Example
Graph a general absolute value function
Sketch y = 2|x + 1| − 3, describing its key features.
Why Does This Work?
|x − h| always reaches its minimum value (zero) exactly at x = h, so a|x − h| does too (when a > 0) — adding k shifts that guaranteed minimum point vertically, giving the exact vertex (h, k), while a independently scales how quickly the graph rises on either side.
Real-Life Example
Manufacturing tolerance cost
A factory's extra cost for producing a part increases the further its size deviates from a target measurement h, with a base cost k and a cost-per-unit-deviation rate a.
Cost = a|size − h| + k is exactly this general absolute value form, letting the factory predict costs for any deviation from the ideal size.
Practice
What is the vertex of y = −3|x − 4| + 2?
HardCommon mistake
Trying to find the vertex by setting x − h = 0 and forgetting to also read off k separately — the full vertex is (h, k), not just the x-value where the absolute value is zero.
Quick Review
- y = a|x − h| + k: vertex (h, k), direction/steepness from a.
- a > 0: opens up (minimum). a < 0: opens down (maximum). |a| > 1: narrower. |a| < 1: wider.
- This general form combines every transformation from sections 6.1-6.3.