If the function y=x2−4

Question

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 #
Remaining Attempts : 3
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Answer 1

To analyze the problem, let's start with the function given:

\[ y = x^2 - 4 \]

Step 1: Find the x-intercepts of the original function

To find the x-intercepts, we set \( y = 0 \):

\[ 0 = x^2 - 4 \]

This simplifies to:

\[ x^2 = 4 \] \[ x = \pm 2 \]

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

Step 2: Reflect the function about the origin

When reflecting a function about the origin, we replace \( (x, y) \) with \( (-x, -y) \). Therefore, the reflected function is:

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

This is equivalent to:

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

Step 3: Find the x-intercepts of the reflected function

To find the x-intercepts of \( y = -x^2 + 4 \), we set \( y \) to 0:

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

This simplifies to:

\[ x^2 = 4 \] \[ x = \pm 2 \]

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

Conclusion

Both the original function and the reflected function have 2 x-intercepts. Therefore, the correct answer to the question regarding the number of x-intercepts of the reflected function is:

Option #3: 2 x-intercepts.