The parabola y = 1 - x² 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 - x².
The parabola y = 1 - x² intersects the x-axis at the points where y = 0, which occurs at x = -1 and x = 1. The equation y = |x| - 1 will also intersect the x-axis at these same points, making it the correct choice.
This equation has x-intercepts at x = 1 and x = 1, so it only intersects the x-axis at one point (x = 1). It does not share both x-intercepts of the parabola, which are at x = -1 and x = 1.
The equation y = |x| has an x-intercept at x = 0. This does not match the x-intercepts of the parabola, which are at x = -1 and x = 1, thereby failing to meet the criteria of having the same x-intercepts.
This equation intersects the x-axis when |x| - 1 = 0, which simplifies to |x| = 1. This results in x = -1 and x = 1, matching the x-intercepts of the parabola y = 1 - x².
The equation y = |x| + 1 does not intersect the x-axis at all because the minimum value of y is 1 (when x = 0). Thus, it lacks any x-intercepts, diverging from the x-intercepts of the parabola.
This equation has an x-intercept at x = -1 but does not include the other x-intercept at x = 1. Therefore, it does not match both x-intercepts of the parabola.
The parabola y = 1 - x² intersects the x-axis at x = -1 and x = 1. Among the given options, only y = |x| - 1 shares both of these x-intercepts, confirming it as the correct choice. The other options either have fewer x-intercepts or do not intersect the x-axis, showcasing the unique characteristics of the selected equation.
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