The Polynomial Shape Lab

Sections 5.1, 5.2 -- Graphs of polynomial and power functions

You have a flat 20 by 30 inch sheet of cardboard and you want to make a box. Cut a square from each corner, fold up the sides. How big should the cut be to get the most volume? Cut too little, you get a flat tray. Cut too much, you get a tall skinny tube. Somewhere in between is the best box, and finding it means understanding the shape of a polynomial.

Step 1 of 8: Power Function Shapes

Four power functions will appear one at a time. Before each draws, predict its shape.

First up: \( y = x \). What does this graph look like?

Step 2 of 8: End Behavior

Here is \( y = x^3 - 100x \) zoomed in. It wiggles, with three turning points. What do you think happens when you zoom way out?

Step 3 of 8: Predicting End Behavior

Predict the end behavior for each polynomial given only the degree and leading coefficient sign. There are four cases.

Step 4 of 8: Turning Points

For each polynomial, predict how many turning points it has before it graphs.

Step 5 of 8: Zeros from Factored Form

\( y = (x - 2)(x + 1)(x - 3) \). Where does this cross the x-axis?

Click all the x-values where you think it crosses. Then hit "Graph it" to see.

Step 6 of 8: Multiplicity, Part One

\( y = (x - 2)^2(x + 1) \). What happens at \(x = 2\)?

Step 7 of 8: Multiplicity, Part Two

\( y = (x - 2)^3(x + 1) \). Now the factor \((x - 2)\) is cubed instead of squared. What happens at \(x = 2\)?

Step 8 of 8: Synthesis

Build me a polynomial: degree 4, positive leading coefficient, zeros at \(x = -3\), \(x = 1\) (multiplicity 2), and \(x = 4\).

Assemble the factors below, then predict the full shape before graphing.

Your polynomial (read-only, built from your factor choices):

\( y = \text{(choose factors below)} \)

Factor 1:

Factor 2 (for multiplicity 2 at x=1):

Factor 3:

Factor 4:

Before you graph: what shape do you predict?

Factor Builder

Build a polynomial from factors and watch the graph update live.

Leading coefficient: 1

Root \(a\) in \((x - a)\): Multiplicity:

Show expanded form

Zoom: 5

Factored form

Degree

End behavior

Zeros

Max turning points

Add factors and adjust the leading coefficient to build a polynomial. The graph updates live.

Try This

The Cardboard Box

Cut \(x\) inches from each corner of a 20 by 30 inch sheet. Fold up the sides.

\( V(x) = x(20 - 2x)(30 - 2x) \)

What is the degree of this polynomial?

What are the zeros?

Explain what each zero means physically.

What is the realistic domain for \(x\)?

Reading a Graph

A graph is shown with x-intercepts at \(x = -2\) (crossing), \(x = 1\) (bouncing), and \(x = 3\) (crossing). Both ends of the graph go up.

Write a possible equation in factored form.

Roller Coaster Design

Design a polynomial for a roller coaster segment: it must start at height 40 at \(x = 0\), touch the ground at \(x = 5\) without going below, rise again, and cross the ground at \(x = 12\).

Build your polynomial using the factor builder below.

Touching the ground at \(x = 5\) without going below means the factor \((x-5)\) appears with even multiplicity. Crossing at \(x = 12\) means odd multiplicity.

Your polynomial:

\( y = a(x-5)^2(x-12) \)

We need \(y(0) = 40\). Plug in \(x = 0\):

\( a(0-5)^2(0-12) = a \cdot 25 \cdot (-12) = -300a = 40 \)

So \( a = -\frac{2}{15} \).

What is the value of \(a\)? (enter a decimal, tolerance within 0.01)

a =

Identify the shapes of power functions and classify them as even or odd.
Determine the end behavior of a polynomial from its degree and leading coefficient.
Find the maximum number of turning points for a polynomial of degree \(n\).
Identify zeros of a polynomial from its factored form.
Describe the behavior of a graph at a zero based on the multiplicity of the corresponding factor.
Construct a polynomial in factored form given zeros, multiplicities, and end behavior.

Quick Check

A polynomial has degree 5 and a negative leading coefficient. It has zeros at \(x = -2\) (multiplicity 1), \(x = 0\) (multiplicity 2), and \(x = 3\) (multiplicity 2). Which statement is true?

Instructor Notes

Teaching Notes

This simulation builds polynomial intuition from the ground up. Steps 1 through 4 establish shape, end behavior, and turning points from power functions. Steps 5 through 7 introduce factored form, zeros, and multiplicity. Step 8 asks students to synthesize everything into one polynomial. The progression is intentional: do not let students skip to the factor builder (Explore) before they have the vocabulary from Discover.

The cardboard box problem in the hook and Starter is a genuine optimization context that returns in calculus. Let students see that finding the best cut is equivalent to finding the peak of the polynomial. They do not need calculus to locate it approximately -- just the graph shape and the domain.

Common Student Errors

Believing the number of turning points always equals \(n - 1\), rather than at most \(n - 1\). Steps 4 addresses this explicitly.
Believing every zero means a crossing. Steps 6 and 7 are designed to break this assumption.
Confusing the sign of the leading coefficient with the direction of the right end. An odd-degree polynomial with a negative leading coefficient goes up on the left and down on the right. Students often reverse this.
Setting factors backward: \((x + 2)\) for a zero at \(x = 2\) instead of \((x - 2)\).

Discussion Questions

Can a degree-3 polynomial have zero turning points? What would that look like?
If you know a polynomial has degree 4 and all real zeros, what is the minimum number of distinct zeros it can have? (Answer: 1, with multiplicity 4.)
The Stretch problem asks why the equation is "possible" and not "the" equation. Push this: what other equations would produce the same graph shape?

Exam Connection

End behavior and multiplicity are frequent exam topics. A common exam question gives a graph and asks for a possible factored form, exactly like the Stretch problem here. Another common format gives a factored form and asks the student to sketch a rough graph, as in Step 8.