Algebra 1 worksheet focusing on identifying key characteristics of quadratic functions from their graphs, including zeros, axis of symmetry, maximum or minimum value, and vertex.

Let's solve each problem step by step using the graph of quadratic functions. We are asked to find:

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 up (minimum) or down (maximum).
4. Vertex – the highest or lowest point on the parabola, located on the axis of symmetry.

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1.

- Graph: Opens downward (U-shaped upside-down), peaks at around $(-2, 4)$, crosses x-axis at $x = -4$ and $x = 0$.
- Zeros: $ x = -4 $, $ x = 0 $
- Axis of symmetry: Midpoint between zeros: $ \frac{-4 + 0}{2} = -2 $ → $ x = -2 $
- Max or Min: Since it opens downward → Maximum
- Vertex: At $ x = -2 $, $ y = 4 $ → $ (-2, 4) $

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

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2.

- Graph: Opens upward, touches x-axis at $x = -2$, so double root.
- Zeros: $ x = -2 $ (double root)
- Axis of symmetry: $ x = -2 $ (since only one zero, it’s the vertex)
- Max or Min: Opens upward → Minimum
- Vertex: $(-2, 0)$

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

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3.

- Graph: Opens upward, crosses x-axis at $x = -2$ and $x = 4$
- Zeros: $ x = -2 $, $ x = 4 $
- Axis of symmetry: $ \frac{-2 + 4}{2} = 1 $ → $ x = 1 $
- Max or Min: Opens upward → Minimum
- Vertex: At $ x = 1 $, $ y = -9 $ → $(1, -9)$

✔ Answer:
- Zeros: $-2, 4$
- Axis of symmetry: $x = 1$
- Max or Min: Min
- Vertex: $(1, -9)$

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4.

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

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

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5.

- Graph: Opens upward, crosses x-axis at $x = 3$ and $x = 7$
- Zeros: $ x = 3 $, $ x = 7 $
- Axis of symmetry: $ \frac{3 + 7}{2} = 5 $ → $ x = 5 $
- Max or Min: Opens upward → Minimum
- Vertex: At $ x = 5 $, $ y = -4 $ → $(5, -4)$

✔ Answer:
- Zeros: $3, 7$
- Axis of symmetry: $x = 5$
- Max or Min: Min
- Vertex: $(5, -4)$

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6.

- Graph: Opens upward, touches x-axis at $x = 2$, so double root.
- Zeros: $ x = 2 $
- Axis of symmetry: $ x = 2 $
- Max or Min: Opens upward → Minimum
- Vertex: $(2, 5)$

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

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7.

- Graph: Opens upward, crosses x-axis at $x = 1$ and $x = 5$
- Zeros: $ x = 1 $, $ x = 5 $
- Axis of symmetry: $ \frac{1 + 5}{2} = 3 $ → $ x = 3 $
- Max or Min: Opens upward → Minimum
- Vertex: At $ x = 3 $, $ y = -4 $ → $(3, -4)$

✔ Answer:
- Zeros: $1, 5$
- Axis of symmetry: $x = 3$
- Max or Min: Min
- Vertex: $(3, -4)$

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8.

- Graph: Opens downward, crosses x-axis at $x = -2$ and $x = 2$
- Zeros: $ x = -2 $, $ x = 2 $
- Axis of symmetry: $ \frac{-2 + 2}{2} = 0 $ → $ x = 0 $
- Max or Min: Opens downward → Maximum
- Vertex: At $ x = 0 $, $ y = 1 $ → $(0, 1)$

✔ Answer:
- Zeros: $-2, 2$
- Axis of symmetry: $x = 0$
- Max or Min: Max
- Vertex: $(0, 1)$

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9.

- Graph: Opens upward, touches x-axis at $x = 4$, so double root.
- Zeros: $ x = 4 $
- Axis of symmetry: $ x = 4 $
- Max or Min: Opens upward → Minimum
- Vertex: $(4, 0)$

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

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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 | -2 | $x = -2$ | Min | $(-2, 0)$ |
| 3 | -2, 4 | $x = 1$ | Min | $(1, -9)$ |
| 4 | -6, -2 | $x = -4$ | Max | $(-4, 2)$ |
| 5 | 3, 7 | $x = 5$ | Min | $(5, -4)$ |
| 6 | 2 | $x = 2$ | Min | $(2, 5)$ |
| 7 | 1, 5 | $x = 3$ | Min | $(3, -4)$ |
| 8 | -2, 2 | $x = 0$ | Max | $(0, 1)$ |
| 9 | 4 | $x = 4$ | Min | $(4, 0)$ |

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🔍 How to Solve These:

- Zeros: Look for where the graph intersects the x-axis.
- Axis of Symmetry: For two distinct zeros: average of the zeros. If only one zero (touching x-axis), it’s the zero itself.
- Max/Min: If the parabola opens upward (U-shape), it has a minimum; if downward, it has a maximum.
- Vertex: The point on the axis of symmetry with the highest or lowest y-value.

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