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.