Absolute Value Functions Worksheet - Free Printable

Let's solve each part of this Absolute Value Functions worksheet step by step.

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🔍 Understanding Absolute Value Functions

The general form of an absolute value function is:

$$
y = a|x - h| + k
$$

Where:
- $(h, k)$ is the vertex.
- If $a > 0$: graph opens up, has a minimum at vertex.
- If $a < 0$: graph opens down, has a maximum at vertex.
- The width compared to parent graph $y = |x|$ depends on $|a|$:
- $|a| > 1$: narrower
- $|a| < 1$: wider
- $|a| = 1$: same width

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## ✔ Part 1: Analyze Each Function

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1) $ y = -|x + 1| $

Rewrite: $ y = -|x - (-1)| + 0 $

- Vertex: $(-1, 0)$
- $a = -1$ → Opens DOWN
- Since it opens down → MAXIMUM at vertex: $0$
- $|a| = 1$ → SAME width as parent

✔ Answer:
- Vertex: $(-1, 0)$
- Opens: DOWN
- MAXIMUM of 0
- SAME

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2) $ y = 7|x - 3| - 4 $

- Vertex: $(3, -4)$
- $a = 7 > 0$ → Opens UP
- Minimum at vertex: $-4$
- $|a| = 7 > 1$ → NARROWER

✔ Answer:
- Vertex: $(3, -4)$
- Opens: UP
- MINIMUM of $-4$
- NARROWER

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3) $ y = -\frac{2}{3}|x - 1| $

- Vertex: $(1, 0)$
- $a = -\frac{2}{3} < 0$ → Opens DOWN
- Maximum at vertex: $0$
- $|a| = \frac{2}{3} < 1$ → WIDER

✔ Answer:
- Vertex: $(1, 0)$
- Opens: DOWN
- MAXIMUM of $0$
- WIDER

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4) $ y = \frac{5}{2}|x + 9| - 1 $

Rewrite: $ y = \frac{5}{2}|x - (-9)| - 1 $

- Vertex: $(-9, -1)$
- $a = \frac{5}{2} > 0$ → Opens UP
- Minimum at vertex: $-1$
- $|a| = \frac{5}{2} = 2.5 > 1$ → NARROWER

✔ Answer:
- Vertex: $(-9, -1)$
- Opens: UP
- MINIMUM of $-1$
- NARROWER

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5) $ y = \frac{3}{4}|x + 3| - 6 $

- Vertex: $(-3, -6)$
- $a = \frac{3}{4} > 0$ → Opens UP
- Minimum at vertex: $-6$
- $|a| = \frac{3}{4} < 1$ → WIDER

✔ Answer:
- Vertex: $(-3, -6)$
- Opens: UP
- MINIMUM of $-6$
- WIDER

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6) $ y = -|x| + 5 $

- Vertex: $(0, 5)$
- $a = -1 < 0$ → Opens DOWN
- Maximum at vertex: $5$
- $|a| = 1$ → SAME

✔ Answer:
- Vertex: $(0, 5)$
- Opens: DOWN
- MAXIMUM of $5$
- SAME

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## ✔ Part 2: Graphing (Steps for Each)

We'll describe how to graph each function. You can plot using key points.

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7) $ y = 3|x - 3| $

- Vertex: $(3, 0)$
- Opens: UP (since $a = 3 > 0$)
- Narrower than parent ($|a| = 3 > 1$)

Plot points:
- $x = 3$: $y = 0$
- $x = 2$: $y = 3|2 - 3| = 3(1) = 3$
- $x = 4$: $y = 3|4 - 3| = 3(1) = 3$
- $x = 1$: $y = 3|1 - 3| = 3(2) = 6$
- $x = 5$: $y = 3|5 - 3| = 6$

→ Draw V-shape with vertex at $(3, 0)$, going up steeply.

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8) $ y = -|x| + 4 $

- Vertex: $(0, 4)$
- Opens: DOWN ($a = -1$)
- Same width

Plot points:
- $x = 0$: $y = 4$
- $x = 1$: $y = -|1| + 4 = 3$
- $x = -1$: $y = -|-1| + 4 = 3$
- $x = 2$: $y = -2 + 4 = 2$
- $x = -2$: same

→ Inverted V, peak at $(0, 4)$

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9) $ y = -|x + 3| + 5 $

Rewrite: $ y = -|x - (-3)| + 5 $

- Vertex: $(-3, 5)$
- Opens: DOWN ($a = -1$)
- Same width

Plot points:
- $x = -3$: $y = 5$
- $x = -2$: $y = -| -2 + 3 | + 5 = -1 + 5 = 4$
- $x = -4$: $y = -| -4 + 3 | + 5 = -1 + 5 = 4$
- $x = -1$: $y = -| -1 + 3 | + 5 = -2 + 5 = 3$
- $x = -5$: $y = -| -5 + 3 | + 5 = -2 + 5 = 3$

→ Inverted V, peak at $(-3, 5)$

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10) $ y = 2|x + 1| - 1 $

- Vertex: $(-1, -1)$
- Opens: UP ($a = 2 > 0$)
- Narrower ($|a| = 2 > 1$)

Plot points:
- $x = -1$: $y = -1$
- $x = 0$: $y = 2|1| - 1 = 2 - 1 = 1$
- $x = -2$: $y = 2| -2 + 1 | - 1 = 2(1) - 1 = 1$
- $x = 1$: $y = 2|2| - 1 = 4 - 1 = 3$
- $x = -3$: $y = 2| -3 + 1 | - 1 = 2(2) - 1 = 3$

→ V-shape, vertex at $(-1, -1)$, steeper than parent

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11) $ y = \frac{4}{3}|x + 2| - 5 $

- Vertex: $(-2, -5)$
- Opens: UP ($a = \frac{4}{3} > 0$)
- Wider? No: $|a| = \frac{4}{3} > 1$ → NARROWER

Wait: $\frac{4}{3} \approx 1.33 > 1$ → NARROWER

Plot points:
- $x = -2$: $y = -5$
- $x = -1$: $y = \frac{4}{3}|1| - 5 = \frac{4}{3} - 5 = -\frac{11}{3} \approx -3.67$
- $x = -3$: $y = \frac{4}{3}| -1 | - 5 = \frac{4}{3} - 5 = -\frac{11}{3} \approx -3.67$
- $x = 0$: $y = \frac{4}{3}|2| - 5 = \frac{8}{3} - 5 = -\frac{7}{3} \approx -2.33$
- $x = -4$: same as $x=0$

→ V-shape, vertex at $(-2, -5)$, slightly narrower than parent

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12) $ y = -\frac{3}{2}|x - 3| + 2 $

- Vertex: $(3, 2)$
- Opens: DOWN ($a = -\frac{3}{2} < 0$)
- $|a| = 1.5 > 1$ → NARROWER

Plot points:
- $x = 3$: $y = 2$
- $x = 2$: $y = -\frac{3}{2}| -1 | + 2 = -\frac{3}{2} + 2 = 0.5$
- $x = 4$: $y = -\frac{3}{2}|1| + 2 = -1.5 + 2 = 0.5$
- $x = 1$: $y = -\frac{3}{2}| -2 | + 2 = -3 + 2 = -1$
- $x = 5$: same as $x=1$

→ Inverted V, peak at $(3, 2)$, steeper than parent

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## ✔ Final Answers Summary

Part 1: Fill in the Blanks

| # | Function | Vertex | Opens | Max/Min | Value | Width |
|---|--------|--------|-------|---------|-------|--------|
| 1 | $ y = -|x+1| $ | $(-1, 0)$ | DOWN | MAXIMUM | 0 | SAME |
| 2 | $ y = 7|x-3| - 4 $ | $(3, -4)$ | UP | MINIMUM | -4 | NARROWER |
| 3 | $ y = -\frac{2}{3}|x-1| $ | $(1, 0)$ | DOWN | MAXIMUM | 0 | WIDER |
| 4 | $ y = \frac{5}{2}|x+9| -1 $ | $(-9, -1)$ | UP | MINIMUM | -1 | NARROWER |
| 5 | $ y = \frac{3}{4}|x+3| -6 $ | $(-3, -6)$ | UP | MINIMUM | -6 | WIDER |
| 6 | $ y = -|x| + 5 $ | $(0, 5)$ | DOWN | MAXIMUM | 5 | SAME |

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Part 2: Graphing Instructions

For each graph:
- Plot the vertex
- Use symmetry and slope based on $a$
- Draw two rays from vertex forming a "V" or inverted "V"

> 💡 Tip: For $y = a|x - h| + k$, the slopes are:
> - Right side: $+a$
> - Left side: $-a$

Example: For $y = 3|x - 3|$, slope is $+3$ to the right, $-3$ to the left.

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Let me know if you'd like these graphs drawn digitally or need printable versions!