Montana Math MT.PLUS.ALG.POLY.3

MathGrades 9–12Polynomial Functions

The Standard

Graph polynomial functions with and without the use of technology, by identifying zeros, relative maxima and minima, and end behavior.

2026 Montana Mathematics Content Standards · Algebraic and Functional Reasoning

Teacher's field guide

What Students Need to Do: Students sketch polynomial graphs by hand and check them with a graphing tool. They locate zeros and high and low points, then use the leading term to predict how both ends of the graph behave.
What Mastery Looks Like: For y=(x+2)(x-1)^2, students mark zeros at -2 and 1 and a high point at (-1,4). They show a low point at (1,0), with the graph crossing at -2 and touching at 1. Their sketch falls to the left and rises to the right.
Common Misconceptions: Students may assume the graph crosses the x-axis at every zero, even when it touches and turns. They may mistake a nearby rise or fall for end behavior, or miss a turning point because the graphing window hides it.
How to Assess It: Ask students to sketch y=(x+2)(x-1)^2 without a calculator. Have them label both zeros, both turning points, and the direction of each end.

Lesson moves

Place sticky notes at (-2,0), (-1,4), and (1,0) on a large coordinate grid, then use string to trace the curve.
Ask why y=(x+2)(x-1)^2 crosses the axis at -2 but touches it at 1; have students explain in writing.
Give pairs four factored polynomials and four graph cards to match, then have them check each match on a graphing calculator.
Sketch V(x)=x(12-2x)(8-2x) for 0<x<4 and estimate which corner cut makes the largest box from a 12-by-8-inch card.

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