Question

If the function y=x2−4
is reflected about the origin, which of the following is true regarding the number of x-intercepts of the reflected function?

Hint: Sketch a graph of the function y=x2−4
, then sketch a graph of the function reflected about the origin.

Option #1: 0 x-intercepts

Option #2: 1 x-intercept

Option #3: 2 x-intercepts

(1 point)
Option #

Answer 1

To determine the number of x-intercepts of the function \( y = x^2 - 4 \) when it is reflected about the origin, we first need to understand the original function and then its reflected counterpart.

1. **Original Function:** The function \( y = x^2 - 4 \) is a parabola that opens upwards. The x-intercepts occur where \( y = 0 \):
\[
x^2 - 4 = 0
\]
\[
x^2 = 4
\]
\[
x = \pm 2
\]
Therefore, the original function has **2 x-intercepts** at \( x = 2 \) and \( x = -2 \).

2. **Reflected Function:** Reflecting about the origin means we replace \( (x, y) \) with \( (-x, -y) \). The reflected function will therefore be:
\[
y = -(x^2 - 4) = -x^2 + 4
\]
This is a downward-opening parabola. To find the x-intercepts of this new function, we again set \( y = 0 \):
\[
-x^2 + 4 = 0
\]
\[
x^2 = 4
\]
\[
x = \pm 2
\]
Thus, the reflected function \( y = -x^2 + 4 \) also has **2 x-intercepts** at \( x = 2 \) and \( x = -2 \).

Now, based on this analysis:

- The reflected function has **2 x-intercepts**.

### Therefore, the correct answer is:
**Option #3: 2 x-intercepts**.