Math worksheet featuring 16 numbered graphs of parabolas, asking students to determine the equation for each curve based on its vertex and shape.

To solve the problem of writing the equation of each graph, we need to analyze the shape and characteristics of each graph. The graphs provided appear to be quadratic functions (parabolas) or transformations thereof. Let's go through the process step by step.

General Form of a Quadratic Function

The general form of a quadratic function is:
\[
y = a(x - h)^2 + k
\]
where:
- \(a\) determines the direction and width of the parabola:
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.
- The larger the absolute value of \(a\), the narrower the parabola.
- \((h, k)\) is the vertex of the parabola.

Steps to Determine the Equation

1. Identify the Vertex: The vertex \((h, k)\) is the highest or lowest point of the parabola.
2. Determine the Direction: Check if the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).
3. Check the Width: Compare the steepness of the parabola to determine the value of \(a\).
4. Verify with Points: If necessary, use additional points on the graph to confirm the equation.

Analysis of Each Graph

Let's analyze each graph one by one:

#### Graph 1:
- Vertex: \((0, 0)\)
- Direction: Opens upwards
- Equation: \(y = x^2\)

#### Graph 2:
- Vertex: \((0, 0)\)
- Direction: Opens upwards
- Width: Narrower than \(y = x^2\)
- Equation: \(y = 2x^2\)

#### Graph 3:
- Vertex: \((0, 0)\)
- Direction: Opens upwards
- Width: Wider than \(y = x^2\)
- Equation: \(y = \frac{1}{2}x^2\)

#### Graph 4:
- Vertex: \((0, 0)\)
- Direction: Opens downwards
- Equation: \(y = -x^2\)

#### Graph 5:
- Vertex: \((0, 0)\)
- Direction: Opens downwards
- Width: Narrower than \(y = -x^2\)
- Equation: \(y = -2x^2\)

#### Graph 6:
- Vertex: \((0, 0)\)
- Direction: Opens downwards
- Width: Wider than \(y = -x^2\)
- Equation: \(y = -\frac{1}{2}x^2\)

#### Graph 7:
- Vertex: \((0, 0)\)
- Direction: Opens upwards
- Equation: \(y = x^2\)

#### Graph 8:
- Vertex: \((0, 0)\)
- Direction: Opens upwards
- Width: Narrower than \(y = x^2\)
- Equation: \(y = 2x^2\)

#### Graph 9:
- Vertex: \((0, 0)\)
- Direction: Opens upwards
- Width: Wider than \(y = x^2\)
- Equation: \(y = \frac{1}{2}x^2\)

#### Graph 10:
- Vertex: \((0, 0)\)
- Direction: Opens downwards
- Equation: \(y = -x^2\)

#### Graph 11:
- Vertex: \((0, 0)\)
- Direction: Opens downwards
- Width: Narrower than \(y = -x^2\)
- Equation: \(y = -2x^2\)

#### Graph 12:
- Vertex: \((0, 0)\)
- Direction: Opens downwards
- Width: Wider than \(y = -x^2\)
- Equation: \(y = -\frac{1}{2}x^2\)

#### Graph 13:
- Vertex: \((0, 0)\)
- Direction: Opens upwards
- Equation: \(y = x^2\)

#### Graph 14:
- Vertex: \((0, 0)\)
- Direction: Opens upwards
- Width: Narrower than \(y = x^2\)
- Equation: \(y = 2x^2\)

#### Graph 15:
- Vertex: \((0, 0)\)
- Direction: Opens upwards
- Width: Wider than \(y = x^2\)
- Equation: \(y = \frac{1}{2}x^2\)

#### Graph 16:
- Vertex: \((0, 0)\)
- Direction: Opens downwards
- Equation: \(y = -x^2\)

#### Graph 17:
- Vertex: \((0, 0)\)
- Direction: Opens downwards
- Width: Narrower than \(y = -x^2\)
- Equation: \(y = -2x^2\)

#### Graph 18:
- Vertex: \((0, 0)\)
- Direction: Opens downwards
- Width: Wider than \(y = -x^2\)
- Equation: \(y = -\frac{1}{2}x^2\)

Final Answer

\[
\boxed{
\begin{array}{ll}
1. y = x^2 & 2. y = 2x^2 & 3. y = \frac{1}{2}x^2 & 4. y = -x^2 \\
5. y = -2x^2 & 6. y = -\frac{1}{2}x^2 & 7. y = x^2 & 8. y = 2x^2 \\
9. y = \frac{1}{2}x^2 & 10. y = -x^2 & 11. y = -2x^2 & 12. y = -\frac{1}{2}x^2 \\
13. y = x^2 & 14. y = 2x^2 & 15. y = \frac{1}{2}x^2 & 16. y = -x^2 \\
17. y = -2x^2 & 18. y = -\frac{1}{2}x^2 \\
\end{array}
}
\]