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 understand the effect of reflecting the function \( y = x^2 - 2 \) about the origin, we first need to determine the function's x-intercepts and then the resulting function after the reflection.

### Step 1: Find the x-intercepts of the original function
To find the x-intercepts of the function \( y = x^2 - 2 \), we set \( y = 0 \):
\[
0 = x^2 - 2
\]
This simplifies to:
\[
x^2 = 2
\]
Taking the square root of both sides gives:
\[
x = \pm \sqrt{2}
\]
Thus, the original function \( y = x^2 - 2 \) has **2 x-intercepts** at \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).

### Step 2: Reflect the function about the origin
Reflecting a function about the origin means replacing \( y \) with \( -y \) and \( x \) with \( -x \). The reflected function can be found by:
\[
y = -(x^2 - 2)
\]
This gives:
\[
y = -x^2 + 2
\]

### Step 3: Find the x-intercepts of the reflected function
Now, we need to determine the x-intercepts of the new function \( y = -x^2 + 2 \). We set \( y = 0 \):
\[
0 = -x^2 + 2
\]
Solving for \( x \) gives:
\[
x^2 = 2
\]
Taking the square root yields:
\[
x = \pm \sqrt{2}
\]
Thus, the reflected function \( y = -x^2 + 2 \) also has **2 x-intercepts** at the same points \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).

### Conclusion
The reflected function about the origin \( y = -x^2 + 2 \) has the same number of x-intercepts as the original function \( y = x^2 - 2 \).

Therefore, the correct statement regarding the number of x-intercepts of the reflected function is:
\[
\text{2 x-intercepts}
\]