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Algebra II

Absolute Value Functions

Lesson

The graph of y=∣x∣y = |x| is a V-shape with its corner at the origin. Every absolute value function is some shifted, stretched, or reflected V.

Vertex form

y=a∣x−h∣+ky = a|x - h| + k
  • Vertex: (h,k)(h, k). Same sign trap as parabolas: (x−h)(x - h) means hh is positive when the term is −h-h.
  • Direction: a>0a > 0 opens UP (vertex is MIN). a<0a < 0 opens DOWN (vertex is MAX).
  • Steepness: ∣a∣|a|. The bigger∣a∣|a|, the narrower the V.

Worked example 1

y=∣x−3∣+5y = |x - 3| + 5

Vertex: (3,5)(3, 5). Opens up. Minimum y is 5.

Worked example 2 — reflection

y=−∣x+1∣+7y = -|x + 1| + 7

Vertex: (−1,7)(-1, 7). Opens DOWN (because a=−1a = -1). Max y is 7.

Worked example 3 — y-intercept

y=2∣x−4∣−1y = 2|x - 4| - 1

At x=0x = 0: 2⋅∣−4∣−1=2⋅4−1=72 \cdot |-4| - 1 = 2 \cdot 4 - 1 = 7.

Family resemblance

Quadratic y=a(x−h)2+ky = a(x-h)^2 + k and absolute value y=a∣x−h∣+ky = a|x-h| + k share the same shifts, reflections, and direction rules. The only difference is the shape: smooth parabola vs sharp V.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Vertex of y=∣x−3∣+5\text{Vertex of } y = |x - 3| + 5

Problem 2

Vertex of y=∣x+2∣−1\text{Vertex of } y = |x + 2| - 1

Problem 3

Vertex of y=∣x∣\text{Vertex of } y = |x|

Problem 4

Vertex of y=∣x−4∣\text{Vertex of } y = |x - 4|

Practice

Standard problems matching the lesson.

Problem 5

Vertex of y=∣x−1∣+4\text{Vertex of } y = |x - 1| + 4

Problem 6

Vertex of y=∣x+5∣−3\text{Vertex of } y = |x + 5| - 3

Problem 7

Vertex of y=−∣x−2∣+8\text{Vertex of } y = -|x - 2| + 8

Problem 8

y-intercept of y=∣x−3∣\text{y-intercept of } y = |x - 3|

Problem 9

y-intercept of y=∣x+4∣−2\text{y-intercept of } y = |x + 4| - 2

Problem 10

y-intercept of y=−∣x−5∣+7\text{y-intercept of } y = -|x - 5| + 7

Problem 11

y-intercept of y=∣x−1∣+6\text{y-intercept of } y = |x - 1| + 6

Problem 12

Value of y=∣x−1∣+2 at x=4\text{Value of } y = |x - 1| + 2 \text{ at } x = 4

Problem 13

Value of y=∣x+5∣−1 at x=−2\text{Value of } y = |x + 5| - 1 \text{ at } x = -2

Problem 14

Value of y=2∣x−3∣ at x=0\text{Value of } y = 2|x - 3| \text{ at } x = 0

Problem 15

y = -|x+1| + 5 opens up (1) or down (0)?

Problem 16

y = |x-7| - 3 opens up (1) or down (0)?

Problem 17

d(x) = |x - 4|. d(7)?

Problem 18

h(t) = -|t-5| + 10. Max?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Vertex of y=−3∣x+1∣+7\text{Vertex of } y = -3|x + 1| + 7

Problem 20

y-intercept of y=2∣x−4∣−1\text{y-intercept of } y = 2|x - 4| - 1

Problem 21

Value of y=∣x−3∣−5 at x=10\text{Value of } y = |x - 3| - 5 \text{ at } x = 10

Problem 22

Vertex of y=0.5∣x−6∣+2\text{Vertex of } y = 0.5|x - 6| + 2

Problem 23

Minimum y of y=∣x−4∣+8\text{Minimum } y \text{ of } y = |x - 4| + 8

Problem 24

Maximum y of y=−∣x+1∣+12\text{Maximum } y \text{ of } y = -|x + 1| + 12

Problem 25

S(x) = -2|x - 3| + 10. S(0)?

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