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










Minimum

no

(0,0)

1

same width

y=-x2










Maximum 

yes

(0,0)

-1

same width

y=4x2










Minimum

no

(0,0)

4

narrower

y=2x2+4









Minimum

no

(0,4)

6

narrower

Based on the given equations, we can complete the table step by step. I'll provide the analysis for each function and fill in the necessary information. Here's how it looks:

| **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)            | 1                      | Same width                       |
| **y = -x²**    | Maximum               | Yes                                  | (0, 0)            | -1                     | Same width                       |
| **y = 4x²**    | Minimum               | No                                   | (0, 0)            | 4                      | Narrower                         |
| **y = 2x² + 4**| Minimum               | No                                   | (0, 4)            | 6                      | Narrower                         |

### Explanation:
1. **Maximum/Minimum**: For a quadratic equation \( y = ax^2 \), if \( a > 0 \), the parabola opens upwards and has a **minimum**; if \( a < 0 \), it opens downwards and has a **maximum**.
   
2. **Reflection in x-axis**: The equation is reflected in the x-axis if there is a negative sign before \( x^2 \), like \( y = -x^2 \).

3. **Turning Point**: The turning point, also called the vertex, is the point where the graph changes direction (minimum or maximum).

4. **y-value when x = 1**: This is the value of the equation when \( x = 1 \).

5. **Wider or Narrower**: The graph of \( y = ax^2 \) is narrower than \( y = x^2 \) when \( |a| > 1 \) and wider when \( 0 < |a| < 1 \).

Would you like to explore any specific details further or need clarifications?

Here are 5 related questions for practice:
1. What is the turning point for \( y = 3x^2 - 5 \)?
2. How would the graph of \( y = \frac{1}{2}x^2 \) compare to \( y = x^2 \)?
3. What happens to the graph if we add a constant \( c \) to the equation \( y = x^2 \)?
4. How do we find the x-value of the turning point in a quadratic equation?
5. How does multiplying the quadratic term by a negative value affect the graph?

**Tip**: To find whether a quadratic equation is wider or narrower than \( y = x^2 \), compare the coefficient of \( x^2 \) with 1.

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

Minimum

no

(0,0)

1

same width

y=-x2

Maximum

yes

(0,0)

-1

same width

y=4x2

Minimum

no

(0,0)

4

narrower

y=2x2+4

Minimum

no

(0,4)

6

narrower

Learn how to graph various quadratic equations like y=x^2, y=-x^2, and y=4x^2, and understand their key properties such as turning points, maximums or minimums, reflections, and how their widths compare to the standard y=x^2. Includes a detailed table and analysis for each equation.

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