Domain, Range, and Axis of Symmetry from Vertex
For any absolute value function f(x) = a|x-h|+k with vertex (h, k), the domain, range, and axis of symmetry follow a fixed pattern — covered in California Reveal Math, Algebra 1 (Grade 9). The domain is always all real numbers (-∞, ∞) because any real x can be substituted. The range depends on the sign of a: if a > 0 the V-shape opens upward and range is [k, ∞); if a < 0 it opens downward and range is (-∞, k]. The axis of symmetry is the vertical line x = h through the vertex. For example, f(x) = 2|x-3|+1 has domain (-∞,∞), range [1,∞), and axis of symmetry x = 3. Reading these features directly from vertex form saves time on exams and builds intuition for graphing.
Key Concepts
For $f(x) = a|x h| + k$ with vertex $(h, k)$:.
Domain: Always $( \infty, \infty)$ Range: $[k, \infty)$ when $a 0$ (opens upward); $( \infty, k]$ when $a < 0$ (opens downward) Axis of Symmetry: $x = h$.
Common Questions
What is the domain of an absolute value function f(x) = a|x-h|+k?
The domain is always all real numbers, (-∞, ∞), because any real number can be substituted for x in an absolute value expression.
What is the range of f(x) = a|x-h|+k?
If a > 0 (opens upward), the range is [k, ∞). If a < 0 (opens downward), the range is (-∞, k]. The vertex y-coordinate k is the boundary.
How do you find the axis of symmetry of an absolute value function?
The axis of symmetry is x = h, the vertical line through the vertex (h, k). It divides the V-shape into two mirror-image halves.
What is the range of f(x) = -4|x+2|+5?
Here a = -4 < 0 and k = 5, so the range is (-∞, 5]. The vertex (−2, 5) is the maximum point.
What is the axis of symmetry for f(x) = |x| - 6?
The vertex is (0, -6), so the axis of symmetry is x = 0 (the y-axis).
Why does a negative value of a change the range of an absolute value function?
When a < 0, the V-shape is flipped downward, so the vertex is the highest point. All output values are at or below k, giving range (-∞, k].
How are domain and range of absolute value functions different?
The domain is always all real numbers. The range is restricted: it is bounded below by k when a > 0, or bounded above by k when a < 0.