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