Lesson 5 — Polynomial Functions: Shape and Key Features

Summary

Learners extend from linear and quadratic functions to polynomials of higher degree, reading a polynomial’s zeros, turning points, and end behavior from its factored form and graph. They model a real relationship — a box’s volume, a projectile’s height — and say what each feature means in context.

Objectives

• Build and interpret polynomial functions that model real relationships and describe their key features. (D05.S2.10.01)

Connection

Cut the corners from a square of cardboard and fold up the sides to make a box. The volume of that box is a polynomial in the side length: too small a cut and the box is shallow, too large and the base is tiny, and somewhere in between the volume is biggest. The shape of that relationship — where it rises, where it turns, where it crosses zero (a box of zero volume) — is exactly what a polynomial function’s key features describe. A thrown ball’s height over time is another polynomial, a parabola. Polynomials are how we describe any smooth curve built from powers of a single quantity, and their features carry real meaning: zeros are “nothing left,” turning points are “the most,” end behavior is “where it goes from here.”

Materials

• Polynomial-features chart
• Graph paper
• Math journal

Preparation

• Copy or draw the polynomial-features chart.
• Retrieval: from Grade 9, linear and quadratic functions and their graphs (D05.S2.09.01, D05.S2.09.02). Today we add degree and look at features that generalize to cubics and beyond.
• Prepare a worked sketch of a factored cubic.

Facilitator note

This lesson is written to the learner (“you”). The idea to land: a polynomial is a sum of powers of x with constant coefficients, and its key features — zeros, turning points, end behavior — can be read from its factored form and carry meaning in a real model. Teach the feature-finding routine explicitly (factor → zeros; degree and leading coefficient → end behavior; sketch → turning points), then let learners practice (S-011).

The intellectual lens: the degree and leading coefficient are a shortcut to the shape, but they can be justified, not memorized. The critical-thinking lens: ask learners to connect each feature to a real question (“when is the volume zero? when is it largest?”). The environment lens: projectile motion — height as a quadratic in time — is a standard, openly licensed model (S-435), and the same shape governs other rise-and-fall processes in nature. The technology lens: a graphing tool confirms a sketch but should not replace the reasoning; read the tool with the math, not instead of it. Preview: Lesson 6 turns to rational functions — ratios of polynomials that model rates and proportions.

Procedure

1. Recall (5 min). From Grade 9: what shape is a quadratic, and what do its x-intercepts tell you? What does the sign of the leading coefficient tell you about a parabola?
2. Meet polynomials (10 min). A polynomial is a sum of terms like aₙxⁿ. Its degree is the highest power; the leading coefficient is the number on that term. You already know degree 1 (linear) and degree 2 (quadratic). Today we add degree 3 (cubic) and 4 (quartic).
3. Read the features (15 min). Worked example: f(x) = (x − 1)(x + 2)(x − 3).
   • Zeros: set each factor to zero → x = 1, −2, 3. These are the x-intercepts.
   • End behavior: the leading term (after multiplying) is x³, so as x → +∞, f(x) → +∞, and as x → −∞, f(x) → −∞ (an odd degree with a positive leading coefficient).
   • Turning points: a cubic has at most two; sketch the curve rising, falling, rising to hit the three zeros in order. Mark each feature on the chart.
4. Model something real (15 min). The volume of a box made by cutting squares of side x from a 10-by-10 sheet is V(x) = x(10 − 2x)². Find: (a) the zeros and what each means (x = 0 and x = 5 → zero volume); (b) the end behavior (degree 3, leading term +4x³ → up on the right and down on the left — but the real box only makes sense on 0 < x < 5, where the base’s side 10 − 2x is positive, so the model is restricted to that interval). Sketch it.
5. Guided practice (5 min). With a partner, sketch g(x) = x²(x − 2) and identify its zeros and end behavior.
6. Close (5 min). Say in one sentence what the zeros, turning points, and end behavior tell you about a real relationship.

Differentiation

• Support: Find zeros of a factored quadratic first, then a factored cubic with integer roots.
• Extension: Sketch a polynomial from its zeros alone (build the factors), and explain why the leading coefficient’s sign sets the far ends.

Assessment

• Formative (peer + self): Can the learner find zeros, describe end behavior, and locate turning points of a factored polynomial, and state what each means in the box model?
• Portfolio artifact (unit): The labeled polynomial-features chart with the box model, added to the functions toolkit.

Home connection

Find a curve around you — an arch, a hill, a ball’s flight. Ask: where does it cross the ground (zeros), where does it peak (turning point), and where does it go far from the middle (end behavior)?

Resources

• On quadratics as models of projectile motion: OpenStax, “Algebra and Trigonometry 2e” (and “College Physics 2e”), https://openstax.org/details/books/algebra-and-trigonometry-2e (S-435).
• On worked examples and guided practice: Kirschner, Sweller & Clark (2006), https://doi.org/10.1207/s15326985ep4102_1 (S-011).