AMC 10 · 2003 · #19

Grade 9 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
transformations-compositionpolynomial-substitutionslope-intercept easier-related-problemconvert-to-algebra ↑ Prerequisites: transformations-composition
📏 Long solution 💡 3 insights
Problem
Take a parabola y=ax²+bx+c and reflect it in the x-axis. Now slide the original parabola 5 units horizontally one way and slide the reflected parabola 5 units horizontally the other way. Describe the graph you get by adding them.

Pick an answer.

(A)
a parabola tangent to the }x\text{-axis
(B)
a parabola not tangent to the }x\text{-axis
(C)
a horizontal line
(D)
a non-horizontal line
(E)
the graph of a cubic function
How to solve
Strategy Introduce a Variable

The question asks for a description of a curve, not a number, so the whole job is to find the shape of f(x)+g(x). First scout with the plainest parabola y=x² (Tool #9) to see what kind of answer to expect cheaply. That scout is only a hint, though: y=x² has b=0 and c=0, and two of the answer choices differ exactly in whether a constant or a slope survives, so a special case cannot decide the question. The real work keeps a, b, c as letters (Tool #4) and adds the two shifted expressions, watching which terms cancel. Because the problem leaves the two sliding directions unassigned, both assignments get listed and checked (Tool #2). Finally the resulting form is matched against the five descriptions and the other four are ruled out (Tool #3).

1STEP 1

Scout with the simplest parabola

The plainest parabola gives 20x, a hint only — it hides the other two coefficients.

f(x)+g(x)=(x+5)²-(x-5)²=20x
2STEP 2

Write f and g with letters

Reflection negates the output and sliding shifts the input, giving p(x+5) and -p(x-5).

f(x)=a(x+5)²+b(x+5)+c, g(x)=-[a(x-5)²+b(x-5)+c]
3STEP 3

Add and watch the cancellations

Adding cancels the constants and every square, leaving 20ax + 10b.

(f+g)(x)=a[(x+5)²-(x-5)²]+b[(x+5)-(x-5)]+(c-c)=20ax+10b
4STEP 4

Check the other direction too

The other direction just flips the sign, so the description does not depend on it.

p(x-5)-p(x+5)=-(20ax+10b)=-20ax-10b
5STEP 5

Name the graph

Since the curve was a parabola the slope is nonzero, so it is a non-horizontal line, choice (D).

y=20ax+10b, slope 20a ≠ 0
Answer
a non-horizontal line
Test the formula on a parabola with all three coefficients alive: a=2, b=3, c=7. Then f(x)=2(x+5)²+3(x+5)+7=2x²+23x+72 and g(x)=-[2(x-5)²+3(x-5)+7]=-2x²+17x-42. Their sum is 40x+30, which is exactly 20ax+10b=20(2)x+10(3). It is a line and its slope 40 is not 0, matching the claim. Notice that c=7 left no trace at all: how high or low the original parabola sits never matters, so no choice of c can flatten the result into a horizontal line. The trap answer is (C): it would need 20a=0, but a=0 would mean the original curve was never a parabola. The answer choices also confirm the reading of the problem — (A) and (B) split on tangency, which only matters if an x² term survives, and it never does.
💡Key takeaway

Reflecting a parabola flips the sign of its x² term and sliding it sideways never touches that term, so when you add the two curves the x² parts wipe out and only a slanted line survives.

  • Scout with the simplest parabola
  • Write f and g with letters
  • Add and watch the cancellations
  • Check the other direction too
  • Name the graph

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