Integrated Math 2 / Unit 5: Modeling with Functions
5.5 Absolute Value Functions
Notes
The absolute value $|x|$ is the distance from 0. Its graph is a V shape.
- $f(x) = a|x - h| + k$ has vertex $(h, k)$. The V opens up if $a > 0$ and down if $a < 0$.
- Same shifts as parabolas: $|x + 2|$ shifts left 2.
- Example: $f(x) = 2|x - 1| + 3$ gives $f(-2) = 2(3) + 3 = 9$.
- The range of $y = -|x| + 4$ is $y \leq 4$.
Practice
What is the vertex of $y = |x + 2| - 5$?
Rewrite as $|x - (-2)| + (-5)$.
If $f(x) = 2|x - 1| + 3$, find $f(-2)$.
$|-2 - 1| = 3$, so $2(3) + 3 = 9$.
What is the range of $y = -|x| + 4$?
The graph opens down with its highest point at $(0, 4)$.
Ready for more? Take the Unit 5 test.