AMC 12 2002a Problem 25

The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y=P(x)$ and $y=Q(x)$ over the interval $-4\leq x\leq4$?

size(760);
defaultpen(linewidth(0.8pt));

void axes(pair O) {
draw((O+(-2.5,0))--(O+(2.5,0)),Arrow);
draw((O+(0,-1.8))--(O+(0,1.8)),Arrow);
draw((O+(-2.2,-.08))--(O+(-2.2,.08)));
draw((O+(-.55,-.08))--(O+(-.55,.08)));
draw((O+(.55,-.08))--(O+(.55,.08)));
draw((O+(2.2,-.08))--(O+(2.2,.08)));
label("$-4$",O+(-2.2,-.28),S);
label("$-1$",O+(-.55,-.28),S);
label("$1$",O+(.55,-.28),S);
label("$4$",O+(2.2,-.28),S);
label("$x$",O+(2.55,-.08),E);
label("$y$",O+(.1,1.8),N);
}

pair A=(-6,3.75), B=(0,3.75), C=(6,3.75), D=(-6,-2.25), E=(0,-2.25);
axes(A); axes(B); axes(C); axes(D); axes(E);
label("$\textbf{(A)}$",A+(-3.05,2.0),E);
label("$\textbf{(B)}$",B+(-3.05,2.0),E);
label("$\textbf{(C)}$",C+(-3.05,2.0),E);
label("$\textbf{(D)}$",D+(-3.05,2.0),E);
label("$\textbf{(E)}$",E+(-3.05,2.0),E);

draw((A+(-1.65,1.9))..(A+(-1.3,.2))..(A+(-.8,-.04))..(A+(-.1,.05))..(A+(.65,-.08))..(A+(1.25,.12))..(A+(1.7,1.95)));
draw((A+(-2.05,.85))..(A+(-1.55,-1.15))..(A+(-.75,-1.35))..(A+(0,-1.25))..(A+(.75,-1.45))..(A+(1.4,-1.15))..(A+(1.95,.9)));

// In (B), the two curves intersect at the tick x=1.
draw((B+(-1.35,1.9))..(B+(-1.12,.35))..(B+(-.82,-.75))..(B+(-.25,-.6))..(B+(.25,-.78))..(B+(.55,-1.05))..(B+(.85,-1.4))..(B+(1.3,.8)));
draw((B+(-.95,-1.8))..(B+(-.7,-.87))..(B+(-.2,-.55))..(B+(.18,-.63))..(B+(.55,-1.05))..(B+(.8,-1.78)));

draw((C+(-1.85,.95))..(C+(-1.25,.4))..(C+(-.65,.18))..(C+(-.1,.3))..(C+(.65,.8))..(C+(1.75,2.0)));
draw((C+(-1.8,-.88))..(C+(-1.35,-.08))..(C+(-.85,.4))..(C+(-.38,.56))..(C+(0,.3))..(C+(.7,-.65))..(C+(1.35,-1.85)));

draw((D+(-2.1,1.2))..(D+(-1.6,1.5))..(D+(-.95,.65))..(D+(-.45,.35))..(D+(.25,.6))..(D+(.8,.85))..(D+(1.3,.45))..(D+(1.8,-.08))..(D+(2.2,.4)));
draw((D+(-2.2,.2))..(D+(-1.7,-1.55))..(D+(-1.15,-1.15))..(D+(-.65,-.75))..(D+(-.15,-1.25))..(D+(.5,-.7))..(D+(.9,-1.0))..(D+(1.35,-1.65))..(D+(1.85,-1.15))..(D+(2.2,.6)));

draw((E+(-1.75,1.55))..(E+(-1.35,.65))..(E+(-.75,.45))..(E+(-.1,.45))..(E+(.6,.3))..(E+(1.25,.4))..(E+(1.75,.9)));
draw((E+(-1.4,2.1))..(E+(-1.1,.6))..(E+(-.8,-.55))..(E+(-.2,-.8))..(E+(.45,-.15))..(E+(.75,0))..(E+(1.1,-.5))..(E+(1.55,-2.0)));