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: \[ \text{2 x-intercepts} \]
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