Difference between revisions of "2003 AMC 12A Problems/Problem 19"
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==Problem== | ==Problem== | ||
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| + | A parabola with equation <math>y=ax^2+bx+c</math> is reflected about the <math>x</math>-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of <math>y=f(x)</math> and <math>y=g(x)</math>, respectively. Which of the following describes the graph of <math>y=(f+g)x</math>? | ||
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| + | <math> \textbf{(A)}\ \text{a parabola tangent to the }x\text{-axis} </math> | ||
| + | <math> \textbf{(B)}\ \text{a parabola not tangent to the }x\text{-axis}\qquad\textbf{(C)}\ \text{a horizontal line} </math> | ||
| + | <math> \textbf{(D)}\ \text{a non-horizontal line}\qquad\textbf{(E)}\ \text{the graph of a cubic function} </math> | ||
==Solution== | ==Solution== | ||
Revision as of 20:27, 1 January 2012
Problem
A parabola with equation 
is reflected about the 
-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of 
and 
, respectively. Which of the following describes the graph of 
?
Solution
If we take the parabola 
and reflect it over the x - axis, we have the parabola 
. Without loss of generality, let us say that the parabola is translated 5 units to the left, and the reflection to the right. Then:
Adding them up produces:
\[(f + g)(x) &= ax^2 + (10a+b)x + 25a + 5b + c - ax^2 + 10ax -bx - 25a + 5b - c &= 20ax + 10b\] (Error compiling LaTeX. Unknown error_msg)
This is a line with slope 
. Since 
cannot be 
(because would be a line) we end up with 
