Key takeaway: An absolute value function contains the absolute value of an expression, such as f(x) = a|x − h| + k, and its graph is a V shape with its corner at (h, k).
Grade 9 – Algebra 1F-IF.C.7bFunctions and Graphs
Definition of Absolute Value Function
An absolute value function uses the distance from zero, which is never negative. The parent function \(f(x) = |x|\) outputs \(x\) when \(x \ge 0\) and \(-x\) when \(x < 0\), so its graph is two straight rays that meet at the origin in a V.
The general form \(f(x) = a|x – h| + k\) moves the corner, called the vertex, to \((h, k)\). The number \(a\) stretches the V when \(|a| > 1\), flattens it when \(|a| < 1\), and flips it upside down when \(a < 0\). The graph is symmetric about the vertical line \(x = h\). An absolute value function models tolerances and distances, such as how far a machine part can be from its target size.
Formula and notation
The absolute value function in transformed form, with its vertex, is
\[f(x) = a\,|x – h| + k, \qquad \text{vertex } (h, k)\]
Each side of the vertex is a line with slope \(a\) or \(-a\). Solve \(f(x) = 0\) by isolating the bars, then split into two equations.
Key facts
- The graph is a V (or an upside-down V) with a sharp corner at the vertex.
- The two arms have slopes of equal size and opposite sign.
- With a negative a, the vertex is the highest point.
- An equation |expression| = c with c > 0 has two cases: expression = c or expression = −c.
Absolute Value Function example
For \(f(x) = -2|x + 1| + 6\), find the vertex, the \(y\)-intercept and the zeros.
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Write \(|x + 1|\) as \(|x – (-1)|\): \(h = -1\), \(k = 6\). The vertex is \((-1, 6)\), and \(a = -2\) makes it a maximum.
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\(f(0) = -2|1| + 6 = 4\), so the \(y\)-intercept is \((0, 4)\).
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Set \(f(x) = 0\): \(|x + 1| = 3\), so \(x + 1 = 3\) or \(x + 1 = -3\), giving \(x = 2\) or \(x = -4\).
Vertex \((-1, 6)\), \(y\)-intercept 4, zeros \(-4\) and \(2\).
Common mistake
Watch out
For \(f(x) = -2|x + 1| + 6\), Yusuf names the vertex \((1, 6)\).
Why it is wrong
The horizontal shift has the opposite sign of the number in the bars: \(x + 1 = x – (-1)\) means \(h = -1\). At \(x = 1\) the function gives \(-2(2) + 6 = 2\), not 6, but at \(x = -1\) it gives \(0 + 6 = 6\), the top of the V.
Related terms and lessons
- Absolute Value
- Parabola
- Piecewise Function
- Transformations of Functions
- Zeros of a Function
- Vertex Form
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