Converting Absolute Value Functions to Piecewise Linear Form
Any absolute value function f(x) = a|x-h|+k can be rewritten as a piecewise linear function — a key skill in California Reveal Math, Algebra 1 (Grade 9). The vertex (h, k) is the split point. For x ≥ h, the expression (x-h) is non-negative so the absolute value is removed directly, giving a(x-h)+k. For x < h, the expression is negative so the absolute value introduces a negation, giving -a(x-h)+k. For example, f(x) = 3|x-2|+1 converts to f(x) = 3x-5 for x≥2 and f(x) = -3x+7 for x<2. This conversion reveals the slopes of both rays (+a and -a), confirms the V-shape, and makes evaluating specific x-values easier.
Key Concepts
Any absolute value function $f(x) = a|x h| + k$ can be rewritten as a piecewise linear function by replacing $|x h|$ with its definition:.
$$f(x) = \begin{cases} a(x h) + k & \text{if } x \geq h \\ a(x h) + k & \text{if } x < h \end{cases}$$.
Common Questions
How do you convert an absolute value function to piecewise form?
For f(x) = a|x-h|+k, split at x=h. Write a(x-h)+k for x≥h and -a(x-h)+k for x<h, then simplify each piece.
Convert f(x) = 3|x-2|+1 to piecewise form.
Split at x=2: f(x) = 3(x-2)+1 = 3x-5 for x≥2, and f(x) = -3(x-2)+1 = -3x+7 for x<2.
Convert f(x) = -2|x+4|+6 to piecewise form.
Rewrite as -2|x-(-4)|+6, split at x=-4: f(x) = -2(x+4)+6 = -2x-2 for x≥-4, and f(x) = 2(x+4)+6 = 2x+14 for x<-4.
Why does the left piece of the piecewise form have a negated slope?
Because |x-h| = -(x-h) when x<h (the expression is negative, and absolute value negates it). Multiplying by a gives -a as the slope of the left ray.
What is the split point when converting an absolute value function?
The split point is always x=h, the x-coordinate of the vertex. This is where the two pieces of the piecewise function meet.
What are the slopes of the two linear pieces of an absolute value function?
The right piece (x≥h) has slope a; the left piece (x<h) has slope -a. They are opposite in sign, forming the V-shape.
Why is piecewise form useful for an absolute value function?
It lets you evaluate the function at specific x-values, solve equations, and confirm graph slopes without needing to reason through the absolute value each time.