Question

If the function y = x² - 2 is reflected about the origin, which of the following is true regarding the number of x-intercepts of the reflected function?

*Sketch the function on the graph, then sketch the function reflected about the origin

0 x-intercepts
1 x-intercept
2 x-intercepts

Answer 1

To analyze the function \( y = x^2 - 2 \) and its reflection about the origin, let's first find the original function's x-intercepts and then determine the properties of the reflected function.

1. **Finding the x-intercepts of \( y = x^2 - 2 \)**:
The x-intercepts occur where \( y = 0 \).

Set the equation equal to zero:
\[
x^2 - 2 = 0
\]
Solving for \( x \):
\[
x^2 = 2
\]
\[
x = \pm \sqrt{2}
\]

Thus, the function \( y = x^2 - 2 \) has **2 x-intercepts** at \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).

2. **Reflecting the function about the origin**:
To reflect a function \( y = f(x) \) about the origin, we transform it to \( y = -f(-x) \).

For our function, this means:
\[
y = -(f(-x)) = -((-x)^2 - 2) = -((x^2) - 2) = -x^2 + 2
\]

The reflected function is \( y = -x^2 + 2 \).

3. **Finding the x-intercepts of \( y = -x^2 + 2 \)**:
Now, we'll set this equation equal to zero to find its x-intercepts:
\[
-x^2 + 2 = 0
\]
Rearranging this gives:
\[
x^2 = 2
\]
\[
x = \pm \sqrt{2}
\]

Thus, the reflected function \( y = -x^2 + 2 \) also has **2 x-intercepts** at \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).

### Conclusion
Since both the original function and the reflected function have **2 x-intercepts**, the answer to the question is:

**2 x-intercepts**.