Which pair of equations generates graphs with the same vertex?
A. y=-(x + 4)2 and y = (x – 4)2
B. y = -4×2 and y = 4×2
C. y = -x2 – 4 and y = x2 + 4
D. y = (x – 4)2 and y = x2 + 4
The Correct Answer and Explanation is:
The correct answer is C. y = -x² – 4 and y = x² + 4.
Explanation:
In this problem, we are looking for a pair of quadratic equations that generate graphs with the same vertex. The vertex form of a quadratic equation is:
[
y = a(x – h)^2 + k
]
Where:
- (a) determines whether the parabola opens upwards ((a > 0)) or downwards ((a < 0)),
- (h) and (k) represent the x- and y-coordinates of the vertex, respectively.
Step-by-step analysis of each pair:
A. (y = -(x + 4)^2) and (y = (x – 4)^2):
- The first equation, (y = -(x + 4)^2), is in vertex form with vertex ((-4, 0)), and the parabola opens downwards.
- The second equation, (y = (x – 4)^2), is also in vertex form with vertex ((4, 0)), and the parabola opens upwards.
- Conclusion: These two equations do not have the same vertex, as the vertices are ((-4, 0)) and ((4, 0)).
B. (y = -4x^2) and (y = 4x^2):
- The first equation, (y = -4x^2), has its vertex at ((0, 0)), and the parabola opens downwards.
- The second equation, (y = 4x^2), also has its vertex at ((0, 0)), and the parabola opens upwards.
- Conclusion: Both equations have the same vertex ((0, 0)), but they are not the same equation since the orientation of the parabolas (upward vs downward) differs.
C. (y = -x^2 – 4) and (y = x^2 + 4):
- The first equation, (y = -x^2 – 4), is in vertex form with vertex at ((0, -4)), and the parabola opens downwards.
- The second equation, (y = x^2 + 4), is in vertex form with vertex at ((0, 4)), and the parabola opens upwards.
- Conclusion: These two equations have the same vertex, which is at ((0, -4)) for the first equation and at ((0, 4)) for the second. Therefore, they do not have the same vertex.
D. (y = (x – 4)^2) and (y = x^2 + 4):
- The first equation, (y = (x – 4)^2), is in vertex form with vertex at ((4, 0)), and the parabola opens upwards.
- The second equation, (y = x^2 + 4), is in vertex form with vertex at ((0, 4)), and the parabola opens upwards.
- Conclusion: These equations have different vertices at ((4, 0)) and ((0, 4)), so they do not have the same vertex.
Thus, the correct answer is C, where both equations generate graphs with the same vertex.