2. 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?
Answer: C
The equation y = |x| - 1 has the same x-intercepts as the parabola y = 1 - x^2.
The x-intercepts of the parabola y = 1 - x^2 occur when y = 0, which leads to the equation 1 - x^2 = 0, giving x-intercepts at x = -1 and x = 1. The equation y = |x| - 1 also has x-intercepts at x = -1 and x = 1, making it the correct choice.
A) y = |x - 1|
This equation has x-intercepts at x = 1 and also at x = 0. Therefore, it does not match the x-intercepts of the parabola, which are at x = -1 and x = 1.
B) y = |x|
The equation y = |x| has an x-intercept only at x = 0. Since the parabola has x-intercepts at x = -1 and x = 1, this option is incorrect.
C) y = |x| - 1
This equation has x-intercepts at x = -1 and x = 1, which are the same as the x-intercepts of the parabola y = 1 - x^2. Thus, this option is correct.
D) y = |x| + 1
The equation y = |x| + 1 does not have any x-intercepts, as it is always positive. This option does not match the x-intercepts of the parabola.
E) y = |x + 1|
This equation has an x-intercept at x = -1 but does not have an x-intercept at x = 1. Therefore, it does not match the x-intercepts of the parabola.
Conclusion
The equation y = |x| - 1 is the only option that correctly matches the x-intercepts of the parabola y = 1 - x^2, which are at x = -1 and x = 1. All other options either fail to meet one or both of these x-intercepts, making them incorrect choices.