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Absolute Value Function

The absolute value function is the function y=∣x∣y = |x|, which outputs the distance of xx from zero. Its graph is a V-shape that opens upward with its vertex at the origin.

The absolute value function f(x)=∣x∣f(x) = |x| is defined as a piecewise function: f(x)=xf(x) = x when x≥0x \geq 0 and f(x)=−xf(x) = -x when x<0x < 0. It is one of the basic parent functions in algebra. The general form f(x)=a∣x−h∣+kf(x) = a|x - h| + k allows for vertical stretches or compressions (controlled by aa), horizontal shifts (controlled by hh), and vertical shifts (controlled by kk), with the vertex located at the point (h,k)(h, k).

Key Formula

f(x)=a∣x−h∣+kf(x) = a|x - h| + k
Where:
  • aa = controls vertical stretch/compression and reflection (if negative, the V opens downward)
  • hh = horizontal shift of the vertex from the origin
  • kk = vertical shift of the vertex from the origin
  • (h,k)(h, k) = the vertex of the absolute value graph

Worked Example

Problem: Graph f(x)=2∣x−3∣−4f(x) = 2|x - 3| - 4 and identify the vertex, direction, and two additional points.
Step 1: Identify the vertex using (h,k)(h, k). Here h=3h = 3 and k=−4k = -4.
Vertex=(3,−4)\text{Vertex} = (3, -4)
Step 2: Determine the direction the V opens. Since a=2>0a = 2 > 0, the graph opens upward. The factor of 2 makes it steeper (narrower) than the parent function y=∣x∣y = |x|.
Step 3: Find a point to the right of the vertex. Plug in x=5x = 5.
f(5)=2∣5−3∣−4=2(2)−4=0f(5) = 2|5 - 3| - 4 = 2(2) - 4 = 0
Step 4: Use symmetry to find a point to the left. The V-shape is symmetric about the vertical line x=h=3x = h = 3. Since (5,0)(5, 0) is 2 units right of the vertex, the mirror point is 2 units left.
Mirror point: (1,0)\text{Mirror point: } (1, 0)
Step 5: Plot the vertex (3,−4)(3, -4) and the two points (1,0)(1, 0) and (5,0)(5, 0), then draw the V-shape through them.
Answer: The graph is a V opening upward with vertex (3,−4)(3, -4), passing through (1,0)(1, 0) and (5,0)(5, 0), and steeper than the parent function by a factor of 2.

Visualization

Why It Matters

Absolute value functions model real situations where only the size of a quantity matters, not its sign. For example, manufacturing tolerances measure how far a part's dimension is from the target — regardless of whether it's too large or too small. Understanding how to shift and stretch the basic V-shape also builds skills you'll reuse with every other family of functions.

Common Mistakes

Mistake: Shifting the graph in the wrong horizontal direction — moving it left when hh is positive.
Correction: In f(x)=a∣x−h∣+kf(x) = a|x - h| + k, the subtraction means the graph shifts right by hh. For example, ∣x−3∣|x - 3| shifts the vertex to x=3x = 3, not x=−3x = -3.
Mistake: Forgetting that a negative value of aa flips the V downward.
Correction: When a<0a < 0, the graph opens downward instead of upward. For instance, y=−∣x∣y = -|x| produces an upside-down V with a maximum at the vertex rather than a minimum.

Related Terms