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All courses Algebra II · F-IF.7.c 32 of 55
Graph polynomial functions using zeros, factorizations, and end behavior.

Inventory polynomial graph features from factored form

Problem
For \(f(x)~=~(x~-~2)(x~+~1)^2\), list each real zero with its multiplicity and crossing or touching behavior, state the degree and leading coefficient, compute the \(y\text{-}\text{intercept}\), and describe \(\text{both}~\text{ends}\).
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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

Read zeros and multiplicities

\[x-2=0~->~x=2,m=1;~x+1=0~->~x=-1,m=2\]

Each factored base gives a zero, and its exponent gives the multiplicity.

02

Translate multiplicity to behavior

\[\text{odd}~m=1~->~\text{crosses}~\text{at}~2;~\text{even}~m=2~->~\text{touches}~\text{at}~-1\]

Odd multiplicity changes sign across a zero, while even multiplicity preserves sign and turns at the axis.

03

Find global algebraic features

\[\text{degree}=1+2=3;~\text{leading}~\text{coefficient}=1;~f(0)=(-2)(1)^2=-2\]

Adding factor degrees gives cubic degree 3. Multiplying leading coefficients gives 1, and direct substitution supplies y-intercept (0,-2).

04

State zeros, intercept, and ends

\[\text{Zeros}/\text{behavior}:~2,~m=1,~\text{crosses};~-1,~m=2,~\text{touches}.~\text{Degree}/\text{leading}~\text{coefficient}:~3~\text{and}~1.~y-\text{intercept}:~-2.~\text{End}~\text{behavior}:~\text{left}~\text{down},~\text{right}~\text{up}.\]

A positive odd-degree leading term falls left and rises right, completing the requested inventory, choice A.

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Another way

  1. Expand only the leading term and evaluate x=0 directly; full expansion is unnecessary.

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Common mistake

Do not confuse multiplicity with the zero value. The exponent 2 belongs to zero -1 and causes touching; zero 2 has multiplicity 1 and crosses.

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

Solution walkthrough video