Graphing Polynomial Functions

Find and analyze zeros of polynomial functions. Grade 12 Math 30-1.

Lesson 2.4 of Polynomial Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Introduction to Factored Form & Multiplicity. Understand factored form and discover how multiplicity determines whether a graph crosses or bounces at each x-intercept. Master the critical even/odd rule for sketching polynomial graphs.
• Introduction to Transformations. Learn how to transform parent functions using parameters a, b, h, and k. Master vertical and horizontal stretches, compressions, reflections, and translations to graph any polynomial quickly.
• Transformations of Cubic Functions. Apply transformations to the cubic parent function f(x) = x³. Work through detailed examples mapping key points, finding intercepts, and determining end behavior.

Inside the lesson — a free preview

Understanding Factored Form

When we factor a polynomial completely, we can express it in a special form that reveals important information about its graph. This is called factored form or factorized form.

General Factored Form:. f(x) = a(x - x_1)(x - x_2)(x - x_3)...(x - x_n)

Why is factored form so powerful?. Factored form immediately tells us WHERE the graph crosses or touches the x-axis. Each factor (x - x_i) corresponds to an x-intercept at x = x_i.

Example:.

Converting Between Forms

Polynomials can be written in different forms:

Standard Form (Expanded):. f(x) = x^3 + 2x^2 - 5x - 6

Factored Form:. f(x) = (x - 1)(x + 2)(x + 3)

Connection:. These are the same polynomial! We can multiply the factored form to get standard form, or factor the standard form to get factored form.

What is Multiplicity?

Multiplicity is one of the most important concepts in understanding polynomial graphs. It tells us HOW a polynomial behaves at each x-intercept.

Definition:. The multiplicity of a root is the number of times that root appears as a factor in the complete factorization of the polynomial.

In simpler terms:. How many times does a particular root repeat?

Why Does Multiplicity Matter?

Multiplicity tells us the behavior of the graph at each x-intercept. Does the graph cross through the axis, or does it just touch and turn around?

The Rule:.

Important:. This is a critical rule for sketching polynomial graphs!

Worked example: s:

Example 1:. f(x) = (x - 2)(x + 3)(x - 5)

Example 2:. f(x) = (x - 2)^2(x + 3)

Example 3:. f(x) = (x + 1)^3(x - 3)^2

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