44. 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?
Answer: C
The equation y = |x| - 1 has the same x-intercepts as the parabola y = 1 - x².
The x-intercepts of the parabola y = 1 - x² are found by setting y to 0, leading to the equation 1 - x² = 0, or x² = 1. This gives x-intercepts at x = 1 and x = -1. The equation y = |x| - 1 also has x-intercepts at these points.
A) y = |x - 1|
This equation has x-intercepts at x = 1 and x = -1 due to the absolute value function. However, it does not intersect the x-axis at x = -1 because |x - 1| = 0 only at x = 1. Thus, this option does not have the same x-intercepts as the parabola.
B) y = |x|
The equation y = |x| has an x-intercept only at x = 0. Since the parabola y = 1 - x² has x-intercepts at x = 1 and x = -1, this option does not share any x-intercepts with the parabola, making it incorrect.
C) y = |x| - 1
This equation has x-intercepts at x = 1 and x = -1, as setting |x| - 1 = 0 results in |x| = 1. This gives the roots x = 1 and x = -1, which are the same x-intercepts as the parabola y = 1 - x², confirming this as the correct answer.
D) y = |x| + 1
The equation y = |x| + 1 does not have any x-intercepts, as the minimum value of |x| + 1 is 1, which is always above the x-axis. Therefore, it does not share any x-intercepts with the parabola, rendering this option incorrect.
E) y = |x + 1|
This equation has an x-intercept at x = -1, but it does not intersect the x-axis at x = 1. Thus, while it has one common x-intercept with the parabola, it fails to meet both, making it incorrect.
Conclusion
The equation y = |x| - 1 is the only option that correctly shares both x-intercepts at x = 1 and x = -1 with the parabola y = 1 - x². All other options either do not share these intercepts or miss one entirely, solidifying option C as the definitive correct answer.