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: $(-1, 0)$
- $a = -1$ → Opens DOWN
- Since it opens down → MAXIMUM at vertex → 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 → 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 → 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 → 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 → 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 → value = 5
- $|a| = 1$ → SAME

✔ 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 (since we can't draw here). Use transformations from the parent graph $y = |x|$:

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

- Vertex: $(3, 0)$
- $a = 3 > 0$ → opens up, narrower
- Transformations:
- Shift right 3 units
- Vertical stretch by factor of 3

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

So plot points: $(3,0), (4,3), (2,3)$ → V-shape pointing up

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

- Vertex: $(0, 4)$
- $a = -1$ → opens down
- Shift up 4 units from $y = -|x|$

Points:
- $x = 0$: $y = 4$
- $x = 1$: $y = -1 + 4 = 3$
- $x = -1$: $y = -1 + 4 = 3$

Plot: $(0,4), (1,3), (-1,3)$ → upside-down V

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

- Vertex: $(-3, 5)$
- $a = -1$ → opens down
- Shift left 3, up 5

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

Plot: $(-3,5), (-2,4), (-4,4)$ → upside-down V centered at $(-3,5)$

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

- Vertex: $(-1, -1)$
- $a = 2 > 0$ → opens up, narrower
- Shift left 1, down 1

Points:
- $x = -1$: $y = -1$
- $x = 0$: $y = 2|1| - 1 = 1$
- $x = -2$: $y = 2| -1| - 1 = 1$

Plot: $(-1,-1), (0,1), (-2,1)$ → narrow V opening up

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

- Vertex: $(-2, -5)$
- $a = \frac{4}{3} > 0$ → opens up, narrower than parent ($>1$)
- Shift left 2, down 5

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 → $y \approx -3.67$

Plot: $(-2,-5), (-1,-3.67), (-3,-3.67)$ → V shape

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

- Vertex: $(3, 2)$
- $a = -\frac{3}{2} < 0$ → opens down, narrower (since $|a| = 1.5 > 1$)
- Shift right 3, up 2

Points:
- $x = 3$: $y = 2$
- $x = 4$: $y = -\frac{3}{2}(1) + 2 = -1.5 + 2 = 0.5$
- $x = 2$: same → $y = 0.5$

Plot: $(3,2), (4,0.5), (2,0.5)$ → upside-down narrow V

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## ✔ Final Answers Summary:

Part 1:

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

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Part 2: Graphs

All graphs should be drawn based on:
- Vertex
- Direction (up/down)
- Width (narrower/wider/same)
- Key points around vertex

You can now sketch them using the instructions above.

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Let me know if you'd like these graphs plotted in text or need help with a specific one!