AMC 10 · 2003 · #20

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
graph-readingpolynomial-rootssystems-of-equations convert-to-algebraformula-substitution ↑ Prerequisites: function-evaluation
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
Part of the graph of the cubic f(x)=ax³+bx²+cx+d is drawn. The picture labels exactly three points: it crosses the x-axis at -1 and at 1, and the y-axis at height 2. Find the coefficient b.

Pick an answer.

(A)
-4
(B)
-2
(C)
0
(D)
2
(E)
4
How to solve
Strategy Convert to Algebra

The picture carries the data, so Tool #1 (Draw a Diagram) comes first as reading rather than drawing: harvest the points whose coordinates the picture states exactly, and ignore the rest of the curve, which is only approximate. Tool #13 (Convert to Algebra) turns each harvested point into an equation by substituting its x-value into ax³+bx²+cx+d. Tool #15 (Organize Information in More Ways) is the finishing move: the three equations look like too little information for four unknowns, but adding the x=1 and x=-1 equations cancels every odd-power term at once and leaves b and d alone.

1STEP 1

Harvest three exact points

Read only the three labelled points; the rest of the curve is not data.

f(-1)=0, f(1)=0, f(0)=2
2STEP 2

Turn each point into an equation

Substituting turns each point into an equation, with signs flipping at the negative input.

f(0)=d=2, f(1)=a+b+c+d=0, f(-1)=-a+b-c+d=0
3STEP 3

Add the two zero equations

Adding the two zero equations cancels the odd powers, leaving b = -d.

(a+b+c+d) + (-a+b-c+d) = 2b+2d = 0 ⟹ b = -d
4STEP 4

Substitute the y-intercept

The y-intercept gives d, so b = -2 — and a and c are never needed, choice (B).

d = 2 ⟹ b = -d = -2 → (B)
Answer
-2
Build a cubic that fits the picture and check b directly. Take a=5/4: then c=-a=-5/4 and f(x)=5/4x³-2x²-5/4x+2=5/4(x+1)(x-1)(x-8/5). It gives f(-1)=0, f(1)=0, f(0)=2, it peaks a little to the left of the y-axis at about 2.17, and it is still below the axis at the right-hand edge of the window because its third root 8/5 lies past the edge — exactly the shape shown. Its x² coefficient is -2, matching. Changing a changes the peak and the third root but never b or d, which is why the picture can leave a unlabelled. The trap is answering 2, choice (D), by reading the y-intercept as b instead of d.
💡Key takeaway

Feeding a polynomial 1 and -1 and adding the results wipes out every odd-power term, leaving only the even ones behind.

  • Harvest three exact points
  • Turn each point into an equation
  • Add the two zero equations
  • Substitute the y-intercept

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