Worksheet for practicing absolute value functions, including identifying key features and graphing.

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 function $y = |x|$ 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
- Has a MAXIMUM at $y = 0$
- $|a| = 1$ → 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
- Has a MINIMUM at $y = -4$
- $|a| = 7 > 1$ → 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
- Has a MAXIMUM at $y = 0$
- $|a| = \frac{2}{3} < 1$ → WIDER

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

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

- Vertex: $(-9, -1)$
- $a = \frac{5}{2} > 0$ → Opens UP
- Has a MINIMUM at $y = -1$
- $|a| = 2.5 > 1$ → 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
- Has a MINIMUM at $y = -6$
- $|a| = 0.75 < 1$ → 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
- Has a MAXIMUM at $y = 5$
- $|a| = 1$ → 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 I can't draw here), but you can plot them using key points.

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

- Vertex: $(3, 0)$
- $a = 3 > 0$ → opens up, narrower
- Use two points:
- $x = 2$: $y = 3|2 - 3| = 3(1) = 3$ → point: $(2, 3)$
- $x = 4$: $y = 3|4 - 3| = 3(1) = 3$ → point: $(4, 3)$
- Draw V-shape with vertex at $(3,0)$ and arms going through those points.

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

- Vertex: $(0, 4)$
- $a = -1$ → opens down, same width
- Points:
- $x = 1$: $y = -|1| + 4 = 3$ → $(1, 3)$
- $x = -1$: $y = -|-1| + 4 = 3$ → $(-1, 3)$
- V-shape opening down from $(0,4)$

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

- Vertex: $(-3, 5)$
- $a = -1$ → opens down, same width
- Points:
- $x = -2$: $y = -|-2 + 3| + 5 = -|1| + 5 = 4$ → $(-2, 4)$
- $x = -4$: $y = -|-4 + 3| + 5 = -| -1 | + 5 = 4$ → $(-4, 4)$
- V-shape down from $(-3,5)$

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

- Vertex: $(-1, -1)$
- $a = 2 > 0$ → opens up, narrower
- Points:
- $x = 0$: $y = 2|0+1| -1 = 2(1) -1 = 1$ → $(0,1)$
- $x = -2$: $y = 2|-2+1| -1 = 2(1) -1 = 1$ → $(-2,1)$
- V-shape up from $(-1,-1)$

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

- Vertex: $(-2, -5)$
- $a = \frac{4}{3} > 0$ → opens up, narrower
- Points:
- $x = -1$: $y = \frac{4}{3}| -1 + 2 | - 5 = \frac{4}{3}(1) - 5 = \frac{4}{3} - 5 = -\frac{11}{3} \approx -3.67$
- $x = -3$: $y = \frac{4}{3}| -3 + 2 | - 5 = \frac{4}{3}(1) - 5 = -\frac{11}{3}$
- Plot vertex and two symmetric points

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

- Vertex: $(3, 2)$
- $a = -\frac{3}{2} < 0$ → opens down, narrower
- Points:
- $x = 2$: $y = -\frac{3}{2}|2-3| + 2 = -\frac{3}{2}(1) + 2 = -1.5 + 2 = 0.5$
- $x = 4$: $y = -\frac{3}{2}|4-3| + 2 = -1.5 + 2 = 0.5$
- V-shape down from $(3,2)$

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## ✔ Final Answers Summary

Part 1: Fill-in-the-blank

| # | 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)
- Use two points on either side (e.g., $x = h \pm 1$, compute $y$)
- Draw V-shape accordingly

> 📌 Tip: Always use symmetry — values equidistant from vertex have same $y$ if $a$ is constant.

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If you'd like, I can generate coordinate tables or ASCII sketches for any of the graphs! Let me know.