Difference between revisions of "1998 AHSME Problems/Problem 14"
(Created page with "== Problem 14 == A parabola has vertex of <math>(4,-5)</math> and has two <math>x-</math>intercepts, one positive, and one negative. If this parabola is the graph of <math>y = ax...") |
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<math> \mathrm{(A) \ } \text{only}\ a \qquad \mathrm{(B) \ } \text{only}\ b \qquad \mathrm{(C) \ } \text{only}\ c \qquad \mathrm{(D) \ } a\ \text{and}\ b\ \text{only} \qquad \mathrm{(E) \ } \text{none}</math> | <math> \mathrm{(A) \ } \text{only}\ a \qquad \mathrm{(B) \ } \text{only}\ b \qquad \mathrm{(C) \ } \text{only}\ c \qquad \mathrm{(D) \ } a\ \text{and}\ b\ \text{only} \qquad \mathrm{(E) \ } \text{none}</math> | ||
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+ | ==Solution== | ||
+ | |||
+ | The vertex of the parabola is at <math>(4,-5)</math>. Since there are two x-intercepts, it must open upwards. If it opened downard, there would be no roots. Thus, <math>a > 0</math>. | ||
+ | |||
+ | The x-coordinate of the vertex is <math>\frac{-b}{2a}</math>. Since <math>a</math> is positive, and the x-intercept is positive, the value <math>-b</math> must be positive too, and <math>b</math> is negative. | ||
+ | |||
+ | By Vieta, the product of the two roots is <math>\frac{c}{a}</math>. Since the two roots are a positive number and a negative number, the product is negative. Since <math>a</math> is positive, that means <math>c</math> must be negative. | ||
+ | |||
+ | Thus <math>\boxed{A}</math> is the right answer - only <math>a</math> is positive. | ||
[[1998 AHSME Problems/Problem 14|Solution]] | [[1998 AHSME Problems/Problem 14|Solution]] |
Revision as of 22:36, 7 August 2011
Problem 14
A parabola has vertex of and has two
intercepts, one positive, and one negative. If this parabola is the graph of
which of
and
must be positive?
Solution
The vertex of the parabola is at . Since there are two x-intercepts, it must open upwards. If it opened downard, there would be no roots. Thus,
.
The x-coordinate of the vertex is . Since
is positive, and the x-intercept is positive, the value
must be positive too, and
is negative.
By Vieta, the product of the two roots is . Since the two roots are a positive number and a negative number, the product is negative. Since
is positive, that means
must be negative.
Thus is the right answer - only
is positive.