Graphing Polynomials: Multiplicity and Sketching

Lesson 2 of 2

In this lesson:

• Cross vs. bounce behavior at zeros (multiplicity)
• Three-step sketching: zeros, end behavior, smooth curve
• Maximum turning points for degree- polynomials

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Define multiplicity in a factored polynomial
2. Predict crossing vs. bouncing at each zero
3. Sketch using zeros, end behavior, and multiplicity
4. State max turning points for degree

Why Do Some Zeros Look Different?

Consider these two graphs near :

• : graph crosses through
• : graph touches at and bounces back

What's different? The exponent on the factor.

That exponent is the multiplicity of the zero.

Multiplicity Determines Crossing or Bouncing

Multiplicity = the exponent on a zero's factor

Higher multiplicity → flatter approach to x-axis

Cross vs. Bounce: Visual Comparison

: crosses | : bounces | : flat cross

Reading Mixed Multiplicities: Full Worked Example

• : mult 2 (even) → bounces
• : mult 1 (odd) → crosses
• : mult 3 (odd) → crosses (flat)
• Degree:

Cross vs. Bounce: Quick Check

At : does the graph cross or bounce?

Identify the multiplicity first, then apply the rule.

Check Your Answer: Both Zeros Cross

• : mult 3 (odd) → crosses
• : mult 1 (odd) → crosses

Both zeros have odd multiplicity — both are crossings.

Sum of Multiplicities Equals the Degree

Degree =

• 6 zeros counting multiplicity; only 3 distinct x-intercepts visible
• Complex roots don't appear on the real-number graph

Bringing It All Together: Three-Step Sketch

To sketch any polynomial from its factored form:

1. Plot the zeros — mark crossing or bouncing at each
2. Draw end-behavior arrows — use degree and leading coefficient
3. Connect smoothly — respect the turning point maximum

Sketch Step-by-Step: Part 1 — Zeros

Step 1 — Find zeros and multiplicities:

• : mult 1 (crosses)
• : mult 2 (bounces)
• : mult 1 (crosses)

Degree:

Sketch Step-by-Step: Part 2 — Connect

Step 2 — End behavior: degree 4 (even), coefficient (negative) → falls both ends

Step 3 — Sketch:

Turning Points: At Most

A degree- polynomial has at most turning points.

Can have fewer — this is a maximum, not an exact count

Guided Practice: Sketch This Polynomial

Your turn — work through the three steps:

1. Find zeros with multiplicities and mark crossing/bouncing
2. Identify degree and end behavior
3. Sketch the rough shape

Try it before the next slide.

Sketch of g(x): Complete Solution

• Zeros: (cross), (bounce), (cross)
• Degree 4, positive → rises both ends
• Sketch: rises left, crosses , bounces , crosses , rises right

Use Technology to Check Your Sketch

Use technology to verify your hand sketch — not to replace it.

• Sketch by hand first (zeros → end behavior → smooth curve)
• Then graph in Desmos or your calculator
• If they differ, check degree, sign, and multiplicities

Practice: Two Polynomials to Sketch

Sketch each polynomial — show zeros, end behavior, and shape:

Work through the three steps for each, then verify with technology.

Practice Sketches: Key Features Confirmed

1. :
zeros (bounce), (cross), (cross); degree 4, positive → rises both ends

2. :
zeros (cross), (cross), (bounce); degree 4, negative → falls both ends

Lesson 2 Summary: Multiplicity and Sketching

✓ Odd multiplicity → crosses; even multiplicity → bounces

✓ Three-step sketch: zeros, end behavior, smooth curve

✓ At most turning points for degree

Watch out: bouncing zeros still count toward the degree

Complete Picture: What You Can Now Do

• Read zeros from factored form (Lesson 1)
• Determine end behavior in two questions (Lesson 1)
• Predict crossing vs. bouncing from multiplicity (Lesson 2)
• Sketch a polynomial from its factored form (Lesson 2)

Next: rational functions — zeros, asymptotes, sign analysis.