Absolute Value Function Vertex Form and Transformations
The vertex form of an absolute value function g(x) = a|x − h| + k identifies all transformation parameters at a glance, as covered in Grade 11 enVision Algebra 1 (Chapter 5: Piecewise Functions). The vertex is located at (h, k) — the corner point where the graph changes direction. The parameter a controls vertical stretch or compression: when |a| > 1 the graph narrows, when 0 < |a| < 1 it widens, and a negative a reflects it across the x-axis creating a downward-opening V-shape.
Key Concepts
The general form of a transformed absolute value function is $g(x) = a|x h| + k$, where the vertex is located at $(h, k)$. The parameter $a$ controls vertical stretch/compression and reflection, $h$ controls horizontal translation, and $k$ controls vertical translation.
Common Questions
What is the vertex form of an absolute value function?
The vertex form is g(x) = a|x − h| + k, where (h, k) is the vertex, a controls vertical stretch/compression and reflection, h is the horizontal translation, and k is the vertical translation.
How do you find the vertex of g(x) = a|x − h| + k?
The vertex is the ordered pair (h, k) — read h directly from the expression inside the absolute value and k from the constant added outside.
What does the parameter a do in an absolute value function?
When |a| > 1, the graph stretches vertically (becomes narrower). When 0 < |a| < 1, it compresses (becomes wider). When a < 0, the graph reflects across the x-axis and opens downward.
How does h affect the graph of g(x) = a|x − h| + k?
A positive h shifts the graph to the right; a negative h shifts it to the left. The vertex moves horizontally to x = h.
How does k affect the graph of g(x) = a|x − h| + k?
A positive k shifts the entire graph up; a negative k shifts it down. The vertex moves vertically to y = k.
What is the axis of symmetry for an absolute value function in vertex form?
The axis of symmetry is the vertical line x = h, passing through the vertex (h, k).