To determine the number of x-intercepts of the function \( y = x^2 - 4 \) when it is reflected about the origin, we first need to understand the original function and then its reflected counterpart.
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Original Function: The function \( y = x^2 - 4 \) is a parabola that opens upwards. The x-intercepts occur where \( y = 0 \): \[ x^2 - 4 = 0 \] \[ x^2 = 4 \] \[ x = \pm 2 \] Therefore, the original function has 2 x-intercepts at \( x = 2 \) and \( x = -2 \).
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Reflected Function: Reflecting about the origin means we replace \( (x, y) \) with \( (-x, -y) \). The reflected function will therefore be: \[ y = -(x^2 - 4) = -x^2 + 4 \] This is a downward-opening parabola. To find the x-intercepts of this new function, we again set \( y = 0 \): \[ -x^2 + 4 = 0 \] \[ x^2 = 4 \] \[ x = \pm 2 \] Thus, the reflected function \( y = -x^2 + 4 \) also has 2 x-intercepts at \( x = 2 \) and \( x = -2 \).
Now, based on this analysis:
- The reflected function has 2 x-intercepts.
Therefore, the correct answer is:
Option #3: 2 x-intercepts.
1 answer