Using the graphing program Desmos generate a graph for the equations below and complete the table:

Equation
Picture
Maximum/ Minimum
Reflected in the x axis (Yes/ No)
Turning point
y value when
 x=1
Wider or narrower than y=x2
y=x2

















y=-x2

















y=4x2

















y=2x2+4




















Let's break down the table and analyze each equation in detail. Below is the table with explanations for each entry:

| Equation      | Maximum/Minimum | Reflected in the x-axis (Yes/No) | Turning Point | y-value when x = 1 | Wider or narrower than y = x² |
|---------------|-----------------|----------------------------------|---------------|--------------------|------------------------------|
| y = x²        | Minimum          | No                               | (0, 0)        | y = 1              | Same (as it's the standard parabola)  |
| y = -x²       | Maximum          | Yes                              | (0, 0)        | y = -1             | Same                          |
| y = 4x²       | Minimum          | No                               | (0, 0)        | y = 4              | Narrower (since 4 > 1, it stretches the parabola vertically) |
| y = 2x² + 4   | Minimum          | No                               | (0, 4)        | y = 6              | Narrower (2 > 1, stretches the parabola, but shifted up by 4 units) |

### Explanation:

1. **Maximum/Minimum**: 
   - The standard parabola \(y = x^2\) opens upwards, so it has a **minimum** point at the vertex.
   - The reflection \(y = -x^2\) opens downwards, so it has a **maximum** point.
   - The parabolas \(y = 4x^2\) and \(y = 2x^2 + 4\) also open upwards, with minimums at their respective turning points.

2. **Reflected in the x-axis**:
   - \(y = -x^2\) is the reflection of \(y = x^2\) across the x-axis, while the other equations are not reflected.

3. **Turning Point**:
   - \(y = x^2\) and \(y = -x^2\) both have their turning points at the origin (0, 0).
   - \(y = 4x^2\) also turns at the origin.
   - \(y = 2x^2 + 4\) has its turning point shifted up by 4 units, so the vertex is at (0, 4).

4. **y-value when x = 1**:
   - For each equation, substitute \(x = 1\) to find the corresponding y-value:
     - \(y = 1^2 = 1\)
     - \(y = -(1^2) = -1\)
     - \(y = 4(1^2) = 4\)
     - \(y = 2(1^2) + 4 = 6\)

5. **Wider or narrower than \(y = x^2\)**:
   - Parabolas with coefficients greater than 1 are **narrower** since they are vertically stretched.
   - \(y = 4x^2\) and \(y = 2x^2 + 4\) are narrower than the standard parabola.

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Would you like further details on graphing these equations or have any specific questions?

### Related Questions:
1. How do you find the turning point of a parabola from its equation?
2. What effect does a negative sign have on the graph of a quadratic equation?
3. How does the coefficient of \(x^2\) influence the width of the parabola?
4. What is the significance of the turning point in the context of maxima and minima?
5. Can a parabola have both a maximum and a minimum point?

### Tip:
To quickly identify whether a parabola opens upwards or downwards, check the sign of the coefficient of \(x^2\). Positive means it opens upwards, negative means it opens downwards.

Using the graphing program Desmos generate a graph for the equations below and complete the table:

Equation
Picture
Maximum/ Minimum
Reflected in the x axis (Yes/ No)
Turning point
y value when x=1
Wider or narrower than y=x2
y=x2
y=-x2
y=4x2
y=2x2+4

Explore how to graph quadratic equations using Desmos and analyze key features such as turning points, reflections, and widths of parabolas. This solution provides detailed insights into equations like y=x², y=-x², y=4x², and y=2x²+4, covering maximum/minimum points and the impact of coefficients.

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