Characteristics of Quadratic Equations Worksheet featuring four parabolas for students to analyze and determine key properties.

To solve the problem, we need to analyze each quadratic function graph provided and determine its properties. Let's go through each graph step by step.

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Graph 1

#### Properties:
1. Domain: The domain of a quadratic function is all real numbers, as there are no restrictions on the input \( x \).
- Domain: \( (-\infty, \infty) \)

2. Range: The graph opens downward, and the vertex is the highest point. The vertex is at \( (0, 4) \), so the maximum value is 4. The range is all \( y \)-values less than or equal to 4.
- Range: \( (-\infty, 4] \)

3. x-intercepts: The graph intersects the \( x \)-axis at \( x = -2 \) and \( x = 2 \).
- x-intercepts: \( (-2, 0) \) and \( (2, 0) \)

4. y-intercept: The graph intersects the \( y \)-axis at \( y = 4 \).
- y-intercept: \( (0, 4) \)

5. Vertex: The vertex is the highest point of the parabola, which is at \( (0, 4) \).
- Vertex: \( (0, 4) \)

6. Maximum value: The maximum value is the \( y \)-coordinate of the vertex.
- Maximum value: 4

7. Axis of symmetry: The axis of symmetry is a vertical line passing through the vertex. Since the vertex is at \( x = 0 \), the axis of symmetry is \( x = 0 \).
- Axis of symmetry: \( x = 0 \)

8. Open up or down: The parabola opens downward.
- Open up or down: Down

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Graph 2

#### Properties:
1. Domain: The domain of a quadratic function is all real numbers.
- Domain: \( (-\infty, \infty) \)

2. Range: The graph opens downward, and the vertex is the highest point. The vertex is at \( (0, 2) \), so the maximum value is 2. The range is all \( y \)-values less than or equal to 2.
- Range: \( (-\infty, 2] \)

3. x-intercepts: The graph intersects the \( x \)-axis at \( x = -4 \) and \( x = 4 \).
- x-intercepts: \( (-4, 0) \) and \( (4, 0) \)

4. y-intercept: The graph intersects the \( y \)-axis at \( y = 2 \).
- y-intercept: \( (0, 2) \)

5. Vertex: The vertex is the highest point of the parabola, which is at \( (0, 2) \).
- Vertex: \( (0, 2) \)

6. Maximum value: The maximum value is the \( y \)-coordinate of the vertex.
- Maximum value: 2

7. Axis of symmetry: The axis of symmetry is a vertical line passing through the vertex. Since the vertex is at \( x = 0 \), the axis of symmetry is \( x = 0 \).
- Axis of symmetry: \( x = 0 \)

8. Open up or down: The parabola opens downward.
- Open up or down: Down

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Graph 3

#### Properties:
1. Domain: The domain of a quadratic function is all real numbers.
- Domain: \( (-\infty, \infty) \)

2. Range: The graph opens upward, and the vertex is the lowest point. The vertex is at \( (1, -1) \), so the minimum value is -1. The range is all \( y \)-values greater than or equal to -1.
- Range: \( [-1, \infty) \)

3. x-intercepts: The graph intersects the \( x \)-axis at \( x = 0 \) and \( x = 2 \).
- x-intercepts: \( (0, 0) \) and \( (2, 0) \)

4. y-intercept: The graph intersects the \( y \)-axis at \( y = 0 \).
- y-intercept: \( (0, 0) \)

5. Vertex: The vertex is the lowest point of the parabola, which is at \( (1, -1) \).
- Vertex: \( (1, -1) \)

6. Minimum value: The minimum value is the \( y \)-coordinate of the vertex.
- Minimum value: -1

7. Axis of symmetry: The axis of symmetry is a vertical line passing through the vertex. Since the vertex is at \( x = 1 \), the axis of symmetry is \( x = 1 \).
- Axis of symmetry: \( x = 1 \)

8. Open up or down: The parabola opens upward.
- Open up or down: Up

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Graph 4

#### Properties:
1. Domain: The domain of a quadratic function is all real numbers.
- Domain: \( (-\infty, \infty) \)

2. Range: The graph opens downward, and the vertex is the highest point. The vertex is at \( (-1, 4) \), so the maximum value is 4. The range is all \( y \)-values less than or equal to 4.
- Range: \( (-\infty, 4] \)

3. x-intercepts: The graph intersects the \( x \)-axis at \( x = -3 \) and \( x = 1 \).
- x-intercepts: \( (-3, 0) \) and \( (1, 0) \)

4. y-intercept: The graph intersects the \( y \)-axis at \( y = 3 \).
- y-intercept: \( (0, 3) \)

5. Vertex: The vertex is the highest point of the parabola, which is at \( (-1, 4) \).
- Vertex: \( (-1, 4) \)

6. Maximum value: The maximum value is the \( y \)-coordinate of the vertex.
- Maximum value: 4

7. Axis of symmetry: The axis of symmetry is a vertical line passing through the vertex. Since the vertex is at \( x = -1 \), the axis of symmetry is \( x = -1 \).
- Axis of symmetry: \( x = -1 \)

8. Open up or down: The parabola opens downward.
- Open up or down: Down

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Final Answer

\[
\boxed{
\begin{array}{|c|c|}
\hline
\text{Graph} & \text{Properties} \\
\hline
1 &
\begin{aligned}
\text{Domain: } & (-\infty, \infty) \\
\text{Range: } & (-\infty, 4] \\
\text{x-intercepts: } & (-2, 0), (2, 0) \\
\text{y-intercept: } & (0, 4) \\
\text{Vertex: } & (0, 4) \\
\text{Maximum value: } & 4 \\
\text{Axis of symmetry: } & x = 0 \\
\text{Open up or down: } & \text{Down} \\
\end{aligned}
\\
\hline
2 &
\begin{aligned}
\text{Domain: } & (-\infty, \infty) \\
\text{Range: } & (-\infty, 2] \\
\text{x-intercepts: } & (-4, 0), (4, 0) \\
\text{y-intercept: } & (0, 2) \\
\text{Vertex: } & (0, 2) \\
\text{Maximum value: } & 2 \\
\text{Axis of symmetry: } & x = 0 \\
\text{Open up or down: } & \text{Down} \\
\end{aligned}
\\
\hline
3 &
\begin{aligned}
\text{Domain: } & (-\infty, \infty) \\
\text{Range: } & [-1, \infty) \\
\text{x-intercepts: } & (0, 0), (2, 0) \\
\text{y-intercept: } & (0, 0) \\
\text{Vertex: } & (1, -1) \\
\text{Minimum value: } & -1 \\
\text{Axis of symmetry: } & x = 1 \\
\text{Open up or down: } & \text{Up} \\
\end{aligned}
\\
\hline
4 &
\begin{aligned}
\text{Domain: } & (-\infty, \infty) \\
\text{Range: } & (-\infty, 4] \\
\text{x-intercepts: } & (-3, 0), (1, 0) \\
\text{y-intercept: } & (0, 3) \\
\text{Vertex: } & (-1, 4) \\
\text{Maximum value: } & 4 \\
\text{Axis of symmetry: } & x = -1 \\
\text{Open up or down: } & \text{Down} \\
\end{aligned}
\\
\hline
\end{array}
}
\]