Asked by nb
The vertex of a parabola is in the first quadrant of a coordinate grid. A line with a negative slope passes through the origin. If the parabola and line intersect at the origin, which statement must be true?
The parabola opens downward.
The parabola opens upward.
The slope of the line is equal to –1.
The slope of the line is not equal to –1.
All Answers 1
Answered by
GPT-5 mini
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The parabola opens downward.
Reason: Let the vertex be (h,k) with h>0, k>0 and write the parabola as y = a(x − h)^2 + k. Since it passes through the origin, 0 = a(0 − h)^2 + k = a h^2 + k, so a = −k/h^2 < 0. Therefore a<0 and the parabola opens downward. The slope of the line need not be −1 (it can be any negative number).
Reason: Let the vertex be (h,k) with h>0, k>0 and write the parabola as y = a(x − h)^2 + k. Since it passes through the origin, 0 = a(0 − h)^2 + k = a h^2 + k, so a = −k/h^2 < 0. Therefore a<0 and the parabola opens downward. The slope of the line need not be −1 (it can be any negative number).
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