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?

Question

*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 analyze the function \( y = x^2 - 2 \) and its reflection about the origin, let's first find the original function's x-intercepts and then determine the properties of the reflected function.

1. Finding the x-intercepts of \( y = x^2 - 2 \): The x-intercepts occur where \( y = 0 \).

   Set the equation equal to zero: \[ x^2 - 2 = 0 \] Solving for \( x \): \[ x^2 = 2 \] \[ x = \pm \sqrt{2} \]

   Thus, the function \( y = x^2 - 2 \) has 2 x-intercepts at \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).

2. Reflecting the function about the origin: To reflect a function \( y = f(x) \) about the origin, we transform it to \( y = -f(-x) \).

   For our function, this means: \[ y = -(f(-x)) = -((-x)^2 - 2) = -((x^2) - 2) = -x^2 + 2 \]

   The reflected function is \( y = -x^2 + 2 \).

3. Finding the x-intercepts of \( y = -x^2 + 2 \): Now, we'll set this equation equal to zero to find its x-intercepts: \[ -x^2 + 2 = 0 \] Rearranging this gives: \[ x^2 = 2 \] \[ x = \pm \sqrt{2} \]

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

Conclusion

Since both the original function and the reflected function have 2 x-intercepts, the answer to the question is:

2 x-intercepts.