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All courses Math III · F-IF.7.c 25 of 55
Graph polynomial functions using zeros, factorizations, and end behavior.

Inventory polynomial graph features from factored form

Problem
For f(x)=(x-2)(x+1)², enter one row per real zero with multiplicity/parity/cross-or-touch behavior, then degree, leading coefficient, y-intercept, and separate left/right end limits.
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Hint

List factor equations and exponents before computing degree and leading coefficient.

Each factor gives a zero and multiplicity; odd multiplicity crosses, even touches. Use the leading term for end behavior and f(0) for y-intercept.

Solution walkthrough

01

Find the zeros

\[x-2~=~0\Rightarrow~x~=~2\qquad~x+1~=~0\Rightarrow~x~=~-1\]

The factored form shows the x-intercepts directly.

02

Use multiplicity for intercept behavior

\[(x-2)^1\Rightarrow~\text{crosses at }2\qquad~(x+1)^2\Rightarrow~\text{touches at }-1\]

Multiplicity \(1\) gives a crossing, while multiplicity \(2\) gives touch-and-turn behavior.

03

Determine the end behavior

\[\text{degree }3~\text{ with positive leading coefficient}\]

A positive cubic falls to the left and rises to the right.

04

State the key graph features

\[\text{zero}~\text{rows}=(x=2,\text{multiplicity}~1,\text{odd},\text{crosses});(x=-1,\text{multiplicity}~2,\text{even},\text{touches});~\text{degree}=3;~\text{leading}~\text{coefficient}=1;~y-\text{intercept}=(0,-2);~\text{left}~\text{end}~\text{as}~x→-∞,f(x)→-∞;~\text{right}~\text{end}~\text{as}~x→∞,f(x)→∞\]

These features tell you how the graph should look at its intercepts and at both ends.

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Another way

  1. You can compare the end behavior to \(x^3\) first, then add the touch-versus-cross behavior from the factors.

!

Common mistake

A common mistake is to say the graph crosses at \(x=-1\), but the squared factor means the graph touches and turns there.

Answer polynomial graph touching at x equals negative 1, crossing at x equals 2, and passing through (0,negative 2), with left-down right-up tails.
degree 3, leading coefficient 1 left down; right up