Difference between revisions of "2001 AMC 12 Problems/Problem 13"
(New page: == Problem == The parabola with equation <math>p(x) = ax^2+bx+c</math> and vertex <math>(h,k)</math> is reflected about the line <math>y=k</math>. This results in the parabola with equatio...) |
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== Solution == | == Solution == | ||
We write <math>p(x)</math> as <math>a(x-h)^2+k</math> (this is possible for any parabola). Then the reflection of <math>p(x)</math> is <math>q(x) = -a(x-h)^2+k</math>. Then we find <math>p(x) + q(x) = 2k</math>. Since <math>p(1) = a+b+c</math> and <math>q(1) = d+e+f</math>, we have <math>a+b+c+d+e+f = 2k</math>, so the answer is <math>\mathrm{E}</math>. | We write <math>p(x)</math> as <math>a(x-h)^2+k</math> (this is possible for any parabola). Then the reflection of <math>p(x)</math> is <math>q(x) = -a(x-h)^2+k</math>. Then we find <math>p(x) + q(x) = 2k</math>. Since <math>p(1) = a+b+c</math> and <math>q(1) = d+e+f</math>, we have <math>a+b+c+d+e+f = 2k</math>, so the answer is <math>\mathrm{E}</math>. | ||
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| + | == See Also == | ||
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| + | {{AMC12 box|year=2001|num-b=12|num-a=14}} | ||
Revision as of 07:09, 15 February 2009
Problem
The parabola with equation 
and vertex 
is reflected about the line 
. This results in the parabola with equation 
. Which of the following equals 
?
Solution
We write 
as 
(this is possible for any parabola). Then the reflection of is 
. Then we find 
. Since 
and 
, we have 
, so the answer is 
.
See Also
| 2001 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 12 |
Followed by Problem 14 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
