The parabola y = 1 - x ^ 2 is shown in the xy-plane above. The graph of which of the following equations has the same x -intercepts as the parabola?

Rationale

y = |x| - 1 has the same x-intercepts as the parabola y = 1 - x^2.

The parabola y = 1 - x^2 intersects the x-axis where y = 0, leading to the equation 1 - x^2 = 0, which simplifies to x = ±1. The equation y = |x| - 1 also has x-intercepts at the same points, as it equals zero at x = ±1.

A (y = |x - 1|) YOUR ANSWER

This equation has x-intercepts at x = 1 and x = -1, but it does not share the same x-intercepts as the given parabola. It only intersects the x-axis at x = 1, while the parabola intersects at x = ±1.

B (y = |x|) YOUR ANSWER

The equation y = |x| has a single x-intercept at x = 0. This is not the same as the x-intercepts of the parabola, which occur at x = ±1. Therefore, this choice does not match.

C (y = |x| - 1) CORRECT

This equation has x-intercepts at the points where |x| - 1 = 0, which simplifies to x = ±1. Thus, this equation shares the same x-intercepts as the parabola y = 1 - x^2.

D (y = |x| + 1) YOUR ANSWER

The equation y = |x| + 1 has no x-intercepts since it is always positive (minimum value is 1). This choice does not match the x-intercepts of the parabola at x = ±1.

E (y = |x + 1|) YOUR ANSWER

This equation has a single x-intercept at x = -1. While it shares one intercept with the parabola, it does not have the second intercept at x = 1, disqualifying it as a match.

Conclusion

Identifying x-intercepts involves determining where the function equals zero. The parabola y = 1 - x^2 has x-intercepts at x = ±1, a characteristic shared by the equation y = |x| - 1. Other options either do not intersect at the required points or lack a second intercept, confirming that choice C is the only equation with the same x-intercepts as the parabola.

The parabola y = 1 - x ^ 2 is shown in the xy-plane above. The graph of which of the following equations has the same x -intercepts as the parabola?

y = |x| - 1 has the same x-intercepts as the parabola y = 1 - x^2.

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