sketch the graph of the quadratic function and compare it with the grapy of y=x squared. g(x)=-x squared+4

To sketch the graph of the quadratic function \( g(x) = -x^2 + 4 \) and compare it with \( y = x^2 \), we can analyze their features:

1. **Shape**: 
   - \( y = x^2 \) is a parabola that opens upwards.
   - \( g(x) = -x^2 + 4 \) is a parabola that opens downwards due to the negative coefficient of \( x^2 \).

2. **Vertex**:
   - The vertex of \( y = x^2 \) is at (0, 0).
   - The vertex of \( g(x) \) is at (0, 4).

3. **Intercepts**:
   - For \( y = x^2 \): 
     - \( y \)-intercept: (0, 0)
     - \( x \)-intercepts: (0, 0)
   - For \( g(x) = -x^2 + 4 \):
     - \( y \)-intercept: (0, 4)
     - \( x \)-intercepts: Solve \( -x^2 + 4 = 0 \) → \( x^2 = 4 \) → \( x = ±2 \) (intercepts at (2, 0) and (-2, 0)).

4. **Direction and Width**:
   - The graph of \( g(x) \) is wider than that of \( y = x^2 \) due to the coefficient of \( x^2 \) being less than 1 in magnitude.

### Sketch

1. **Graph of \( y = x^2 \)**: A standard upward-opening parabola with vertex at (0, 0).
2. **Graph of \( g(x) = -x^2 + 4 \)**: A downward-opening parabola with vertex at (0, 4) and intercepts at (2, 0) and (-2, 0).

### Comparison

- \( g(x) = -x^2 + 4 \) is a reflection of \( y = x^2 \) across the horizontal line \( y = 2 \) (the line through its vertex).

If you need a visual graph or further analysis, please let me know!

### Questions for Further Exploration

1. How do transformations affect the graph of a quadratic function?
2. What are the implications of the vertex form of a quadratic function?
3. How can we determine the maximum or minimum value of a quadratic function?
4. What role do the coefficients play in determining the shape of a parabola?
5. Can you find the axis of symmetry for both quadratic functions?

**Tip**: Always identify the vertex and intercepts when sketching quadratic functions, as they provide key points for the graph.

Sketch the graph of the quadratic function and compare it with the graph of y=x squared. g(x)=-x squared+4

Explore the graphs of the quadratic functions g(x) = -x^2 + 4 and y = x^2. Learn about their vertices, intercepts, and shapes, and how they compare. This guide is perfect for students in Grades 7-10.

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