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Medium

Graphing y = |x − h| + k

Simple Explanation

The graph of y = |x − h| + k is a V-shape opening upward, with its sharp corner (the vertex) located exactly at (h, k). Compared to the basic graph y = |x|, it is shifted h units horizontally and k units vertically.

Why Do We Need It?

Absolute value functions model any situation with a sharp minimum and two symmetric, straight-line sides — recognizing the shifted vertex form lets you sketch one instantly, without plotting many points.

See It

Graph of y = |x − 2| + 1
-4-4-2-2224466880xy(2, 1)

An upward-opening V-shaped graph with its vertex at (2, 1)

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

Find the vertex and sketch

Find the vertex of y = |x − 3| + 2, and state a point on each side of the vertex.

    Why Does This Work?

    |x − h| is always zero exactly when x = h (its smallest possible value, since absolute value is never negative), and grows equally in both directions as x moves away from h — so adding k simply shifts that guaranteed minimum point up or down to become the vertex (h, k).

    Real-Life Example

    Minimum cost from a central warehouse

    A delivery cost depends on distance from a single warehouse at location h, with a fixed base cost k.

    Cost = |location − h| + k is exactly this absolute value form — cost is lowest (k) right at the warehouse and rises equally in both directions as distance increases.

    Practice

    What is the vertex of y = |x + 4| − 1?

    Medium

    Common mistake

    Reading h with the wrong sign — in y = |x + 4| − 1, h is −4 (not 4), because the form is |x − h|, and x + 4 = x − (−4).

    Quick Review

    • y = |x − h| + k is a V-shape opening upward with vertex (h, k).
    • The graph is symmetric about the vertical line x = h.
    • Watch the sign: |x + 4| means h = −4, not h = 4.
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