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

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==Problem==
 
==Problem==
  
==Solution==
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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>
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<math> \textbf{(B)}\ \text{a parabola not tangent to the }x\text{-axis}\qquad\textbf{(C)}\ \text{a horizontal line} </math>
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<math> \textbf{(D)}\ \text{a non-horizontal line}\qquad\textbf{(E)}\ \text{the graph of a cubic function} </math>
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==Solution 1==
  
 
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:
 
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:
 
   
 
   
<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>  
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<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>  
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Adding them up produces:
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<cmath> \begin{align*} (f + g)(x) = ax^2 + (10a+b)x + 25a + 5b + c - ax^2 + 10ax -bx - 25a + 5b - c = 20ax + 10b \end{align*}</cmath>
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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{(D)} \text{ a non-horizontal line }}</math>
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==Solution 2: less computation==
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WLOG let the parabola be <math>y=x^2</math>, and the reflected parabola is thus <math>y=-x^2</math>. We can also assume <cmath>f(x) = (x+5)^2 = x^2 + 10x + 25</cmath> and <cmath>g(x) = -(x-5)^2 = -x^2 + 10x - 25.</cmath> Therefore, <cmath>(f+g)(x) = x^2 + 10x + 25 - x^2 + 10x - 25 = 20x,</cmath> which is a non-horizontal line <math>\Rightarrow \boxed{\textbf{D}}</math>.
  
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>
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-MP8148
  
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>
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==See Also==
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{{AMC12 box|year=2003|ab=A|num-b=18|num-a=20}}
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{{MAA Notice}}

Latest revision as of 22:55, 4 January 2019

Problem

A parabola with equation $y=ax^2+bx+c$ is reflected about the $x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $y=f(x)$ and $y=g(x)$, respectively. Which of the following describes the graph of $y=(f+g)(x)?$

$\textbf{(A)}\ \text{a parabola tangent to the }x\text{-axis}$ $\textbf{(B)}\ \text{a parabola not tangent to the }x\text{-axis}\qquad\textbf{(C)}\ \text{a horizontal line}$ $\textbf{(D)}\ \text{a non-horizontal line}\qquad\textbf{(E)}\ \text{the graph of a cubic function}$

Solution 1

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:

\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*}

Adding them up produces: \begin{align*} (f + g)(x) = ax^2 + (10a+b)x + 25a + 5b + c - ax^2 + 10ax -bx - 25a + 5b - c = 20ax + 10b \end{align*}

This is a line with slope $20a$. Since $a$ cannot be $0$ (because $ax^2 + bx + c$ would be a line) we end up with $\boxed{\textbf{(D)} \text{ a non-horizontal line }}$

Solution 2: less computation

WLOG let the parabola be $y=x^2$, and the reflected parabola is thus $y=-x^2$. We can also assume \[f(x) = (x+5)^2 = x^2 + 10x + 25\] and \[g(x) = -(x-5)^2 = -x^2 + 10x - 25.\] Therefore, \[(f+g)(x) = x^2 + 10x + 25 - x^2 + 10x - 25 = 20x,\] which is a non-horizontal line $\Rightarrow \boxed{\textbf{D}}$.

-MP8148

See Also

2003 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
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

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