# Difference between revisions of "2003 AMC 12A Problems/Problem 19"

Epicfailiure (talk | contribs) (Created page with '==Problem== ==Solution== If we take the parabola ax^2 + bx + c and reflect it over the x - axis, we have the parabola -ax^2 - bx - c. Without loss of generality, let us say tha…') |
(→Solution) |
||

Line 3: | Line 3: | ||

==Solution== | ==Solution== | ||

− | If we take the parabola ax^2 + bx + c and reflect it over the x - axis, we have the parabola -ax^2 - bx - c. Without loss of generality, let us say that the parabola is translated 5 units to the left, and the reflection to the right. Then: | + | If we take the parabola <math>ax^2 + bx + c</math> and reflect it over the x - axis, we have the parabola <math>-ax^2 - bx - c</math>. Without loss of generality, let us say that the parabola is translated 5 units to the left, and the reflection to the right. Then: |

− | f(x) = a(x+5)^2 + b(x+5) | + | <cmath> \begin{align*} f(x) &= a(x+5)^2 + b(x+5) + c = ax^2 + (10a+b)x + 25a + 5b + c \\ g(x) &= -a(x-5)^2 - b(x-5) - c = -ax^2 + 10ax -bx - 25a + 5b - c \end{align*} </cmath> |

− | g(x) = -a(x-5)^2 - b(x-5) - c = -ax^2 + 10ax -bx - 25a + 5b - c | ||

− | Adding them up: | + | Adding them up produces: <cmath> (f + g)(x) &= ax^2 + (10a+b)x + 25a + 5b + c - ax^2 + 10ax -bx - 25a + 5b - c &= 20ax + 10b </cmath> |

− | + | This is a line with slope <math>20a</math>. Since <math>a</math> cannot be <math>0</math> (because <math>ax^2 + bx + c</math> would be a line) we end up with <math>\boxed{\textbf{(C)} \text{ a horizontal line }}</math> |

## Revision as of 19:21, 7 August 2011

## Problem

## 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. ! Misplaced alignment tab character &.)

This is a line with slope . Since cannot be (because would be a line) we end up with