Absolute Value Functions Worksheet - Free Printable

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

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🔍 General Form of Absolute Value Function:

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

Where:
- Vertex: $(h, k)$
- If $a > 0$: graph opens UP, has a MINIMUM at the vertex
- If $a < 0$: graph opens DOWN, has a MAXIMUM at the vertex
- The width compared to the parent function $y = |x|$ depends on $|a|$:
- $|a| > 1$: NARROWER
- $0 < |a| < 1$: WIDER
- $|a| = 1$: SAME

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## ✔ Part 1: Analyze Each Function

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

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

- Vertex: $(-1, 0)$
- $a = -1 < 0$ → Opens DOWN
- Since it opens down, it has a MAXIMUM at vertex → Max value: 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 → Min value: -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 → Max value: 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 → Min value: -1
- $|a| = 2.5 > 1$ → 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 → Min value: -6
- $|a| = 0.75 < 1$ → 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 → Max value: 5
- $|a| = 1$ → SAME width

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

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## ✔ Part 2: Graph Each Function

We'll describe how to graph each one (you can draw them on grid paper). We’ll use key points and transformations.

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

- Vertex: $(3, 0)$
- $a = 3 > 0$ → Opens UP
- Narrower than parent ($|a| > 1$)
- Use points:
- $x = 3$: $y = 0$
- $x = 4$: $y = 3|1| = 3$
- $x = 2$: $y = 3| -1| = 3$
- $x = 5$: $y = 3|2| = 6$
- $x = 1$: $y = 3| -2| = 6$

Plot: V-shape with vertex at (3,0), steeper than $|x|$

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

- Vertex: $(0, 4)$
- $a = -1$ → Opens DOWN
- Same width
- Points:
- $x = 0$: $y = 4$
- $x = 1$: $y = -1 + 4 = 3$
- $x = -1$: $y = -1 + 4 = 3$
- $x = 2$: $y = -2 + 4 = 2$
- $x = -2$: $y = 2$

Plot: 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
- Same width
- Points:
- $x = -3$: $y = 5$
- $x = -2$: $y = -|1| + 5 = 4$
- $x = -4$: $y = -| -1| + 5 = 4$
- $x = -1$: $y = -|2| + 5 = 3$
- $x = -5$: $y = -| -2| + 5 = 3$

Plot: Inverted V centered at (-3,5)

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

- Vertex: $(-1, -1)$
- $a = 2 > 0$ → Opens UP
- Narrower
- Points:
- $x = -1$: $y = -1$
- $x = 0$: $y = 2|1| - 1 = 1$
- $x = -2$: $y = 2| -1| - 1 = 1$
- $x = 1$: $y = 2|2| - 1 = 3$
- $x = -3$: $y = 2| -2| - 1 = 3$

Plot: Steeper V, vertex at (-1,-1)

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

- Vertex: $(-2, -5)$
- $a = \frac{4}{3} > 0$ → Opens UP
- $|a| > 1$ → NARROWER
- Points:
- $x = -2$: $y = -5$
- $x = -1$: $y = \frac{4}{3}|1| - 5 = \frac{4}{3} - 5 = -\frac{11}{3} \approx -3.67$
- $x = -3$: same as above → $-\frac{11}{3}$
- $x = 0$: $y = \frac{4}{3}|2| - 5 = \frac{8}{3} - 5 = -\frac{7}{3} \approx -2.33$
- $x = -4$: same → $-\frac{7}{3}$

Plot: V-shape, narrow, vertex at (-2,-5)

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

- Vertex: $(3, 2)$
- $a = -\frac{3}{2} < 0$ → Opens DOWN
- $|a| = 1.5 > 1$ → NARROWER
- Points:
- $x = 3$: $y = 2$
- $x = 4$: $y = -\frac{3}{2}(1) + 2 = -1.5 + 2 = 0.5$
- $x = 2$: same → $0.5$
- $x = 5$: $y = -\frac{3}{2}(2) + 2 = -3 + 2 = -1$
- $x = 1$: same → $-1$

Plot: Inverted V, narrower, peak at (3,2)

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## ✔ Final Answers Summary:

Part 1:

| Problem | Vertex | Opens | Max/Min | Value | Width |
|--------|--------|--------|---------|-------|--------|
| 1 | (-1, 0) | DOWN | MAXIMUM | 0 | SAME |
| 2 | (3, -4) | UP | MINIMUM | -4 | NARROWER |
| 3 | (1, 0) | DOWN | MAXIMUM | 0 | WIDER |
| 4 | (-9, -1) | UP | MINIMUM | -1 | NARROWER |
| 5 | (-3, -6) | UP | MINIMUM | -6 | WIDER |
| 6 | (0, 5) | DOWN | MAXIMUM | 5 | SAME |

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Part 2: Graphing Instructions

For each graph:
- Plot the vertex
- Determine direction (up/down) and width (narrower/wider/same)
- Plot 2–3 additional points on each side of the vertex
- Draw a V-shaped graph accordingly

> 📌 You should now be able to neatly sketch all six graphs based on these instructions.

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Let me know if you'd like a visual sketch or printable version!