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MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

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3.2 Polynomial Functions and Their Graphs

Connect factored form to zeros.

Before you solve

Ideas in this lesson

Explore the lesson brief and the worked example for Idea 11. Create a free account to continue with the remaining ideas and practice.

01Recognize polynomials and permissible graph shapesWorked steps · five questions02Transform and sketch power functionsWorked steps · five questions03Explain end behavior using leading terms and windowsWorked steps · five questions04Factor and sketch polynomials using sign tablesWorked steps · five questions05Use multiplicity to describe local zero behaviorWorked steps · five questions06Apply and limit the intermediate value theoremWorked steps · five questions07Locate all numerical extrema and interpret domain and rangeWorked steps · five questions08Reason about possible extrema counts and minimum degreeWorked steps · five questions09Compare parameterized polynomial familiesWorked steps · five questions10Find and verify intersections of polynomial graphsWorked steps · five questions11Prove polynomial parity and decompose mixed polynomialsWorked steps · five questions12Compare powers and construct large-power tablesWorked steps · five questions13Analyze polynomial profit and population modelsWorked steps · five questions14Optimize open-box and edge-constrained box volumesWorked steps · five questions15Model the strength of a beam cut from a circular logWorked steps · five questions16Optimize a shelter with specified faces onlyWorked steps · five questions
Lesson brief and rules

3.2 Polynomial Functions and Their Graphs

A zero is an xx-value where the function equals zero. A factor (x−a)(x - a) gives a zero at aa.

Remember
Repeated factors have repeated zeros.
Worked example 11
01 / First state the general form and exponent restrictions for a polynomial. Classify 2x3+3x−72 x^{3} + 3 x - 7, x+1\sqrt{x} + 1 and 2+1x22 + \dfrac{1}{x^{2}} as polynomials or not. For the polynomial give its degree, leading coefficient and constant term. Show your work; a teacher must review the construction or explanation. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
02 / The form is anxn+an−1xn−1+⋯+a0a_{\mathrm{nx}}^{n}+a_{n-1}x^{n-1}+\cdots+a_0, where nn is a nonnegative integer and an≠0a_n\ne0. All exponents are nonnegative integers.
03 / A polynomial has only nonnegative integer exponents after simplification.
04 / A square root corresponds to exponent one half; a reciprocal power has a negative exponent.
05 / Read the highest exponent and its nonzero coefficient from the valid expression.
Worked example 2 · Recognize polynomials and permissible graph shapes

A proposed graph is continuous but has a sharp corner at (−1,2)(-1,2). Another proposed graph has a hole at (3,−1)(3,-1). Could either be a polynomial graph? Explain whether being smooth and continuous alone proves a graph is polynomial. Show your work; a teacher must review the construction or explanation. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

Polynomial functions are defined at every real input and are continuous.
Their graphs are smooth, so a corner is forbidden and a hole is forbidden.
A smooth continuous curve such as an exponential is not a polynomial; the graph properties alone do not prove polynomial form.
Written-work review:

Read the graphs

Explain end behavior using leading terms and windows

Also sketch and label all four combinations of odd and even degree and positive and negative leading coefficient. Determine the end behavior of P(x)=x3+2x2−3x+1P(x)=x^{3} + 2 x^{2} - 3 x + 1. Graph it together with its leading term in a small and a large window, and use a ratio table at inputs −10,10-10, 10 and 100100 to explain the comparison. Show your work; a teacher must review the construction or explanation. Identify which of the following candidate curves A–DD can represent PP, and justify your match with its tail directions.
Match a polynomial to a candidate graphAxes labeled x and y. Window x from -3 to 3, y from -35 to 35. Open circles exclude endpoints; filled circles include them.-3-2-10123-27-18-9091827A: solidAB: dashedBC: dottedCD: dash-dotDxy
Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

For positive leading coefficient, even degree has both tails up and odd degree has left down and right up. For negative leading coefficient, even degree has both tails down and odd degree has left up and right down.
The leading term is x3x^{3}.
Dividing by the leading term gives P(x)x3=x3+2x2−3x+1x3\dfrac{P(x)}{x^{3}}=\dfrac{x^{3} + 2 x^{2} - 3 x + 1}{x^{3}}.
Every remaining negative power tends to zero as the magnitude of the input grows.
Thus the ratio approaches 11; lower terms may still visibly matter in a small window.
Small-window reference:

Full polynomial and leading termAxes labeled x and y. Window x from -2 to 2, y from -9.52 to 12.52. Open circles exclude endpoints; filled circles include them.-2-1012-9-6-3036912P: solidPQ: dashedQxy

Large-window reference:
Full polynomial and leading termAxes labeled x and y. Window x from -10 to 10, y from -1173.68 to 1344.68. Open circles exclude endpoints; filled circles include them.-10-8-6-4-20246810-945-630-31503156309451260P: solidPQ: dashedQxy

Only A has the positive odd-degree tail pattern required by this expression; B reverses those tails, C has both ends up, and D has both ends down.
Four tail-pattern references:
Four possible polynomial tail patternsAxes labeled x and y. Window x from -2 to 2, y from -17 to 17. Open circles exclude endpoints; filled circles include them.-2-1012-15-10-5051015odd +: solidodd +odd -: dashedodd -even +: dottedeven +even -: dash-doteven -xy

Written-work review:
  • Show the requested algebra or numerical evidence, not merely a selected option.
  • Include all requested coordinates, domains, units, comparisons and conclusions.
  • For a graph, label axes, intercepts and extrema; use correct end behavior and a suitable scale.

Answer: For positive leading coefficient, even degree has both tails up and odd degree has left down and right up. For negative leading coefficient, even degree has both tails down and odd degree has left up and right down. Left end: negative infinity; right end: positive infinity. Ratios −10-10: 0.769000,100.769000, 10: 1.171000,1001.171000, 100: 1.0197011.019701. Candidate A matches: left end down and right end up.

Locate all numerical extrema and interpret domain and range

Define a local maximum and a local minimum, distinguishing them from global extrema. Read the displayed graph of P(x)=x3−3xP(x)=x^{3} - 3 x.
Graph of fAxes labeled x and y. Window x from -3.732050807568877 to 3.732050807568877, y from -7.0 to 7.0. Open circles exclude endpoints; filled circles include them.-3-2-10123-6-4-20246f: solidxy
Estimate all local extrema and intercepts, then give domain, range, and left and right end behavior. Show your work; a teacher must review the construction or explanation. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

A local maximum is at least as high as nearby values; a local minimum is at least as low as nearby values. A global extremum compares with the entire domain, not just a neighborhood.
Begin with a window containing all displayed zeros and extrema, then expand both axes to check the tails.
The reference turning points are maximum at (−1.00,2.00)(-1.00, 2.00); minimum at (1.00,−2.00)(1.00, -2.00).
Round only after locating the points numerically.
The leading term is x3x^{3}. Left end: negative infinity; right end: positive infinity.
An odd-degree polynomial has opposite infinite tails; an even-degree polynomial uses the lowest or highest turning value as its global range boundary.
Reference sketch (( use the exact coordinates in the written solution )):

Reference graphAxes labeled x and y. Window x from -2.732050807568877 to 2.732050807568877, y from -16.586767294881014 to 16.586767294881014. Open circles exclude endpoints; filled circles include them.-2-1012-15-10-5051015model: solidxy

Written-work review:
  • Show the requested algebra or numerical evidence, not merely a selected option.
  • Include all requested coordinates, domains, units, comparisons and conclusions.
  • For a graph, label axes, intercepts and extrema; use correct end behavior and a suitable scale.

Answer: A local maximum is at least as high as nearby values; a local minimum is at least as low as nearby values. A global extremum compares with the entire domain, not just a neighborhood. maximum at (−1.00,2.00)(-1.00, 2.00); minimum at (1.00,−2.00)(1.00, -2.00); zeros −1.73,0.00,1.73-1.73, 0.00, 1.73; yy-intercept (0,0)(0,0); domain all real numbers; range all real numbers. Left end: negative infinity; right end: positive infinity.

Compare powers and construct large-power tables

Compare y=x6y=x^{6} and y=x7y=x^{7} on [−1,1][-1,1]. Make a table at −1,−0.5,0,0.5-1, -0.5, 0, 0.5 and 11, sketch both curves, and explain how still larger even and odd powers behave. Also order x2,x3,x4,x5x^{2},x^{3},x^{4},x^{5} at x=12x=\dfrac12 and x=2x=2. Show your work; a teacher must review the construction or explanation. Also graph y=xcy=x^{c} for c=1,3,5,7c=1,3,5,7 together on one coordinate system and describe the effect of the exponent. Finally identify curves A,BA, B and CC in the supplied graph as x2,x4,x6x^{2},x^{4},x^{6}.
Identify the power curvesAxes labeled x and y. Window x from -1.2 to 1.2, y from 0 to 3. Open circles exclude endpoints; filled circles include them.-1010123A: solidAB: dashedBC: dottedCxy
Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

For absolute inputs below 11, repeated multiplication reduces magnitude.
At 11 all powers remain 11; at −1-1 parity determines the sign.
Even powers are nonnegative and symmetric; odd powers retain the sign and origin symmetry.
For positive inputs above 11, higher powers grow faster rather than becoming smaller.
Reference plot on the unit interval:

Even and odd powersAxes labeled x and y. Window x from -1 to 1, y from -1.1 to 1.1. Open circles exclude endpoints; filled circles include them.-101-101even power: solideven powerodd power: dashedodd powerxy

On the positive unit interval, higher powers lie lower, while beyond 1 their order reverses. All three even curves pass through (0,0)(0,0) and (1,1)(1,1).
Odd-exponent family reference:
Odd power familyAxes labeled x and y. Window x from -1.2 to 1.2, y from -4 to 4. Open circles exclude endpoints; filled circles include them.-101-4-3-2-101234c = 1: solidc = 1c = 3: dashedc = 3c = 5: dottedc = 5c = 7: dash-dotc = 7xy

Written-work review:
  • Show the requested algebra or numerical evidence, not merely a selected option.
  • Include all requested coordinates, domains, units, comparisons and conclusions.
  • For a graph, label axes, intercepts and extrema; use correct end behavior and a suitable scale.

Answer: Table (( even, odd )): −1-1: 11, −1-1; −12- \dfrac{1}{2}: 164\dfrac{1}{64}, −1128- \dfrac{1}{128}; 00: 00, 00; 12\dfrac{1}{2}: 164\dfrac{1}{64}, 1128\dfrac{1}{128}; 11: 11, 11. At one half: x5<x4<x3<x2x^{5}<x^{4}<x^{3}<x^{2}; at 22: x2<x3<x4<x5x^{2}<x^{3}<x^{4}<x^{5}. For the supplied graph, AA is x2x^{2}, BB is x4x^{4} and CC is x6x^{6}.

15 questions with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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