Parabola Review Worksheet | mrmillermath - Free Printable

To solve the problem, we need to analyze each parabola and identify its characteristics: vertex, minimum/maximum value, domain, range, and zeros. Let's go through each parabola step by step.

Parabola 1:

#### Graph Description:
- The parabola opens upwards.
- The vertex is at the origin \((0, 0)\).
- The parabola intersects the x-axis at \((0, 0)\).

#### Characteristics:
1. Vertex: \((0, 0)\)
2. Min/Max: Minimum value is \(0\) (since it opens upwards).
3. Domain: All real numbers, \((-∞, ∞)\).
4. Range: \([0, ∞)\) (since the parabola starts at \(y = 0\) and goes up).
5. Zeros: \(x = 0\) (the parabola touches the x-axis at the origin).

Parabola 2:

#### Graph Description:
- The parabola opens downwards.
- The vertex is at approximately \((2, 4)\).
- The parabola intersects the x-axis at two points: \((0, 0)\) and \((4, 0)\).

#### Characteristics:
1. Vertex: \((2, 4)\)
2. Min/Max: Maximum value is \(4\) (since it opens downwards).
3. Domain: All real numbers, \((-∞, ∞)\).
4. Range: \((−∞, 4]\) (since the parabola starts at \(y = 4\) and goes down).
5. Zeros: \(x = 0\) and \(x = 4\) (the parabola crosses the x-axis at these points).

Parabola 3:

#### Graph Description:
- The parabola opens downwards.
- The vertex is at approximately \((0, 4)\).
- The parabola intersects the x-axis at two points: \((-2, 0)\) and \((2, 0)\).

#### Characteristics:
1. Vertex: \((0, 4)\)
2. Min/Max: Maximum value is \(4\) (since it opens downwards).
3. Domain: All real numbers, \((-∞, ∞)\).
4. Range: \((−∞, 4]\) (since the parabola starts at \(y = 4\) and goes down).
5. Zeros: \(x = -2\) and \(x = 2\) (the parabola crosses the x-axis at these points).

Parabola 4:

#### Graph Description:
- The parabola opens upwards.
- The vertex is at approximately \((1, -1)\).
- The parabola does not intersect the x-axis.

#### Characteristics:
1. Vertex: \((1, -1)\)
2. Min/Max: Minimum value is \(-1\) (since it opens upwards).
3. Domain: All real numbers, \((-∞, ∞)\).
4. Range: \([-1, ∞)\) (since the parabola starts at \(y = -1\) and goes up).
5. Zeros: None (the parabola does not touch or cross the x-axis).

Final Answers:

1. Parabola 1:
- Vertex: \((0, 0)\)
- Min/Max: Minimum value is \(0\)
- Domain: \((-∞, ∞)\)
- Range: \([0, ∞)\)
- Zeros: \(x = 0\)

2. Parabola 2:
- Vertex: \((2, 4)\)
- Min/Max: Maximum value is \(4\)
- Domain: \((-∞, ∞)\)
- Range: \((−∞, 4]\)
- Zeros: \(x = 0\) and \(x = 4\)

3. Parabola 3:
- Vertex: \((0, 4)\)
- Min/Max: Maximum value is \(4\)
- Domain: \((-∞, ∞)\)
- Range: \((−∞, 4]\)
- Zeros: \(x = -2\) and \(x = 2\)

4. Parabola 4:
- Vertex: \((1, -1)\)
- Min/Max: Minimum value is \(-1\)
- Domain: \((-∞, ∞)\)
- Range: \([-1, ∞)\)
- Zeros: None

Boxed Final Answer:

\[
\boxed{
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Parabola} & \text{Vertex} & \text{Min/Max} & \text{Domain} & \text{Range} & \text{Zeros} \\
\hline
1 & (0, 0) & \text{Minimum: } 0 & (-\infty, \infty) & [0, \infty) & x = 0 \\
\hline
2 & (2, 4) & \text{Maximum: } 4 & (-\infty, \infty) & (-\infty, 4] & x = 0, x = 4 \\
\hline
3 & (0, 4) & \text{Maximum: } 4 & (-\infty, \infty) & (-\infty, 4] & x = -2, x = 2 \\
\hline
4 & (1, -1) & \text{Minimum: } -1 & (-\infty, \infty) & [-1, \infty) & \text{None} \\
\hline
\end{array}
}
\]