Let's analyze the function \( y = -x^2 + 7 \) step by step and determine its domain, range, x-intercepts, and y-intercept.

### 1. **Domain**:
The domain of a quadratic function is the set of all possible x-values. Since this is a basic quadratic function, there are no restrictions on \( x \). Therefore, the domain is:

\[
\text{Domain} = (-\infty, \infty)
\]

### 2. **Range**:
The range of a quadratic function depends on its vertex and whether it opens upwards or downwards. Since the coefficient of \( x^2 \) is negative (\( -1 \)), the parabola opens downwards, and the maximum value is at the vertex.

To find the vertex, we use the formula for the x-coordinate of the vertex in a quadratic function, \( x = -\frac{b}{2a} \). In the equation \( y = -x^2 + 7 \), we have \( a = -1 \) and \( b = 0 \), so the x-coordinate of the vertex is:

\[
x = -\frac{0}{2(-1)} = 0
\]

Substituting \( x = 0 \) into the equation for \( y \), we get the y-coordinate of the vertex:

\[
y = -(0)^2 + 7 = 7
\]

Since the parabola opens downwards, the maximum value of \( y \) is 7, and the range is:

\[
\text{Range} = (-\infty, 7]
\]

### 3. **y-intercept**:
To find the y-intercept, set \( x = 0 \) and solve for \( y \):

\[
y = -(0)^2 + 7 = 7
\]

So, the y-intercept is \( (0, 7) \).

### 4. **x-intercepts**:
To find the x-intercepts, set \( y = 0 \) and solve for \( x \):

\[
0 = -x^2 + 7
\]
\[
x^2 = 7
\]
\[
x = \pm \sqrt{7}
\]

So, the x-intercepts are \( (\sqrt{7}, 0) \) and \( (-\sqrt{7}, 0) \), which are approximately \( (2.65, 0) \) and \( (-2.65, 0) \).

### Graph:
The graph is a downward-opening parabola with vertex \( (0, 7) \), and it crosses the x-axis at approximately \( x = 2.65 \) and \( x = -2.65 \).

Would you like a detailed graph of this function? 

### Relative Questions:
1. How does the vertex form of a quadratic equation help in graphing?
2. Why does the parabola open downwards in this equation?
3. How can you find the x-intercepts for any quadratic function?
4. What happens to the range of a parabola if the parabola opens upwards?
5. What would be the effect of shifting this parabola vertically by 3 units?

### Tip:
To graph any quadratic function, finding the vertex and intercepts gives you key points that make sketching the graph easier.

Explore the quadratic function y = -x^2 + 7 with a detailed explanation of its domain, range, and intercepts. Learn how to find the vertex, x-intercepts, and y-intercept of this downward-opening parabola. Suitable for Grades 9-10 algebra students.

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