Question

If the function y=x^2 -4 is reflected about the origiwhich of the following is tru regarding the number of x-intercepts of the reflected function
Options 0 x- intercepts
1 x- intercept
2 x- intercepts

Answer 1

To analyze the number of x-intercepts of the function \( y = x^2 - 4 \) when it is reflected about the origin, we first determine the x-intercepts of the original function.

The x-intercepts occur where \( y = 0 \):

\[
x^2 - 4 = 0
\]

Solving this equation:

\[
x^2 = 4
\]
\[
x = 2 \quad \text{or} \quad x = -2
\]

Thus, the function \( y = x^2 - 4 \) has 2 x-intercepts at \( (2, 0) \) and \( (-2, 0) \).

Next, we consider the reflection of this function about the origin. The reflection about the origin is given by replacing \( x \) with \( -x \) and \( y \) with \( -y \). So, for the function \( y = x^2 - 4 \), its reflection is:

\[
-y = (-x)^2 - 4
\]

Simplifying gives:

\[
-y = x^2 - 4
\]
\[
y = -x^2 + 4
\]

Now we need to find the x-intercepts of the new function \( y = -x^2 + 4 \). Setting this equal to zero to find the x-intercepts:

\[
-x^2 + 4 = 0
\]

Solving for \( x \):

\[
-x^2 = -4
\]
\[
x^2 = 4
\]
\[
x = 2 \quad \text{or} \quad x = -2
\]

Thus, the reflected function \( y = -x^2 + 4 \) also has 2 x-intercepts at \( (2, 0) \) and \( (-2, 0) \).

Therefore, the number of x-intercepts of the reflected function is:

**2 x-intercepts**