Algebra 1 worksheet focusing on the characteristics of quadratic functions, including zeros, axis of symmetry, max or min value, and vertex, with nine different parabolic graphs for analysis.

Let's solve each problem step by step based on the quadratic function graphs provided. For each graph, we need to determine:

1. Zeros: The x-values where the graph crosses the x-axis (i.e., $ y = 0 $).
2. Axis of symmetry: The vertical line that divides the parabola into two symmetric halves. It passes through the vertex.
3. Max or Min: Whether the parabola opens upward (minimum) or downward (maximum).
4. Vertex: The highest or lowest point on the parabola.

We’ll go through each one.

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1.

- Graph: Parabola opening downward, vertex at approximately $ (-2, 4) $
- Zeros: Where it crosses x-axis → $ x = -4 $ and $ x = 0 $
- Axis of symmetry: Midpoint between zeros: $ \frac{-4 + 0}{2} = -2 $ → $ x = -2 $
- Max or Min: Opens downward → Maximum
- Vertex: $ (-2, 4) $

✔ Answer:
- Zeros: $ -4, 0 $
- Axis of symmetry: $ x = -2 $
- Max or Min: Max
- Vertex: $ (-2, 4) $

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2.

- Graph: Parabola opening upward, touches x-axis at $ x = -1 $
- Zeros: Only one zero (touches axis) → $ x = -1 $ (double root)
- Axis of symmetry: $ x = -1 $
- Max or Min: Opens upward → Minimum
- Vertex: $ (-1, 0) $

✔ Answer:
- Zeros: $ -1 $
- Axis of symmetry: $ x = -1 $
- Max or Min: Min
- Vertex: $ (-1, 0) $

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3.

- Graph: Parabola opening upward, vertex at $ (1, -8) $
- Zeros: Crosses x-axis at $ x = -1 $ and $ x = 3 $
- Axis of symmetry: Midpoint: $ \frac{-1 + 3}{2} = 1 $ → $ x = 1 $
- Max or Min: Opens upward → Minimum
- Vertex: $ (1, -8) $

✔ Answer:
- Zeros: $ -1, 3 $
- Axis of symmetry: $ x = 1 $
- Max or Min: Min
- Vertex: $ (1, -8) $

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4.

- Graph: Parabola opening downward, vertex at $ (-4, 2) $
- Zeros: Crosses x-axis at $ x = -6 $ and $ x = -2 $
- Axis of symmetry: Midpoint: $ \frac{-6 + (-2)}{2} = -4 $ → $ x = -4 $
- Max or Min: Opens downward → Maximum
- Vertex: $ (-4, 2) $

✔ Answer:
- Zeros: $ -6, -2 $
- Axis of symmetry: $ x = -4 $
- Max or Min: Max
- Vertex: $ (-4, 2) $

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5.

- Graph: Parabola opening upward, vertex at $ (5, -4) $
- Zeros: Crosses x-axis at $ x = 2 $ and $ x = 8 $
- Axis of symmetry: Midpoint: $ \frac{2 + 8}{2} = 5 $ → $ x = 5 $
- Max or Min: Opens upward → Minimum
- Vertex: $ (5, -4) $

✔ Answer:
- Zeros: $ 2, 8 $
- Axis of symmetry: $ x = 5 $
- Max or Min: Min
- Vertex: $ (5, -4) $

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6.

- Graph: Parabola opening upward, vertex at $ (2, 5) $
- Zeros: Does not cross x-axis → No real zeros
- Axis of symmetry: Vertical line through vertex → $ x = 2 $
- Max or Min: Opens upward → Minimum
- Vertex: $ (2, 5) $

✔ Answer:
- Zeros: None
- Axis of symmetry: $ x = 2 $
- Max or Min: Min
- Vertex: $ (2, 5) $

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7.

- Graph: Parabola opening upward, vertex at $ (2, -3) $
- Zeros: Crosses x-axis at $ x = 0 $ and $ x = 4 $
- Axis of symmetry: Midpoint: $ \frac{0 + 4}{2} = 2 $ → $ x = 2 $
- Max or Min: Opens upward → Minimum
- Vertex: $ (2, -3) $

✔ Answer:
- Zeros: $ 0, 4 $
- Axis of symmetry: $ x = 2 $
- Max or Min: Min
- Vertex: $ (2, -3) $

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8.

- Graph: Parabola opening downward, vertex at $ (0, 2) $
- Zeros: Crosses x-axis at $ x = -2 $ and $ x = 2 $
- Axis of symmetry: Midpoint: $ \frac{-2 + 2}{2} = 0 $ → $ x = 0 $
- Max or Min: Opens downward → Maximum
- Vertex: $ (0, 2) $

✔ Answer:
- Zeros: $ -2, 2 $
- Axis of symmetry: $ x = 0 $
- Max or Min: Max
- Vertex: $ (0, 2) $

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9.

- Graph: Parabola opening upward, vertex at $ (4, -2) $
- Zeros: Crosses x-axis at $ x = 2 $ and $ x = 6 $
- Axis of symmetry: Midpoint: $ \frac{2 + 6}{2} = 4 $ → $ x = 4 $
- Max or Min: Opens upward → Minimum
- Vertex: $ (4, -2) $

✔ Answer:
- Zeros: $ 2, 6 $
- Axis of symmetry: $ x = 4 $
- Max or Min: Min
- Vertex: $ (4, -2) $

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✔ Final Answers Summary:

| Problem | Zeros | Axis of Symmetry | Max or Min | Vertex |
|--------|-------|------------------|------------|--------|
| 1 | -4, 0 | $ x = -2 $ | Max | (-2, 4) |
| 2 | -1 | $ x = -1 $ | Min | (-1, 0) |
| 3 | -1, 3 | $ x = 1 $ | Min | (1, -8) |
| 4 | -6, -2 | $ x = -4 $ | Max | (-4, 2) |
| 5 | 2, 8 | $ x = 5 $ | Min | (5, -4) |
| 6 | None | $ x = 2 $ | Min | (2, 5) |
| 7 | 0, 4 | $ x = 2 $ | Min | (2, -3) |
| 8 | -2, 2 | $ x = 0 $ | Max | (0, 2) |
| 9 | 2, 6 | $ x = 4 $ | Min | (4, -2) |

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