2006 AMC 10A Problems/Problem 8
Problem
A parabola with equation passes through the points (2,3) and (4,3). What is
?
Solution
Solution 1
Substitute the points (2,3) and (4,3) into the given equation for (x,y).
Then we get a system of two equations:
Subtracting the first equation from the second we have:
Then using in the first equation:
is the answer.
Solution 2
Alternatively, notice that since the equation is that of a monic parabola, the vertex is likely . Thus, the form of the equation of the parabola is
. Expanding this out, we find that
.
Solution 3
The points given have the same -value, so the vertex lies on the line
.
The -coordinate of the vertex is also equal to
, so set this equal to
and solve for
, given that
:
Now the equation is of the form . Now plug in the point
and solve for
:
See also
2006 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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All AMC 10 Problems and Solutions |