Graphing y = −|x − h| + k
Simple Explanation
The graph of y = −|x − h| + k is an upside-down V, opening downward, with its highest point (the vertex) at (h, k). The negative sign in front of the absolute value flips the graph vertically.
Why Do We Need It?
Just like with parabolas, recognizing whether a and its sign points the graph up or down tells you immediately whether the vertex is the lowest or highest value the function reaches.
See It
A downward-opening upside-down V graph with its vertex (highest point) at (2, 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
Find the vertex of a downward V
Find the vertex of y = −|x − 5| + 4, and state whether it is a maximum or minimum.
Why Does This Work?
Multiplying |x − h| (which is always ≥ 0) by a negative number makes every value it produces ≤ 0, so subtracting from k instead of adding to it — the graph reaches its highest value exactly at the vertex, and only decreases moving away from it.
Real-Life Example
Signal strength near a transmitter
A radio signal's strength is strongest directly at the transmitter's location and decreases steadily and symmetrically with distance.
This is modeled by a downward absolute value graph — maximum signal strength (the vertex) right at the transmitter, falling off equally in both directions.
Practice
The graph of y = −|x − 1| + 6 has a vertex at (1, 6). Is this a maximum or a minimum?
MediumCommon mistake
Applying the negative sign to h or k instead of the whole absolute value term — the minus sign in −|x−h|+k flips the V's direction; it does not change the vertex's coordinates (h, k) at all.
Quick Review
- y = −|x − h| + k opens downward, with a maximum vertex at (h, k).
- The negative sign flips the graph vertically but leaves the vertex location unchanged.
- Sign of a: positive → minimum (opens up); negative → maximum (opens down) — same rule as quadratics.