Lesson 2.18
Sketching Polynomial Graphs
This is the capstone lesson — you'll combine end behavior, zeros with multiplicity, turning points, and a few test points to sketch a complete polynomial graph by hand.
Introduction
You've learned every individual tool throughout Unit 2. Now it's time to put them all together. This systematic checklist produces accurate polynomial graphs every time, without a calculator.
Past Knowledge
End behavior, zeros, multiplicity, turning points — everything from Chapters 5-8.
Today's Goal
Follow a 5-step process to sketch any polynomial graph accurately by hand.
Future Success
Sketching graphs connects directly to analyzing rational functions in Unit 3 and modeling in calculus.
Key Concepts
The 5-Step Sketching Checklist
End Behavior
Use the leading coefficient and degree to determine where the graph starts and ends.
Find Zeros
Factor completely to find all x-intercepts.
Determine Multiplicity
For each zero, decide: cross (odd) or bounce (even).
Find the y-intercept
Evaluate f(0) to get the point where the graph crosses the y-axis.
Plot Test Points & Connect
Choose x-values between zeros, evaluate, plot all points, and draw a smooth curve.
End Behavior Cheat Sheet
Even + Positive
↗ ↗
Both ends up
Even + Negative
↘ ↘
Both ends down
Odd + Positive
↘ ↗
Down-left, up-right
Odd + Negative
↗ ↘
Up-left, down-right
Worked Examples
Example 1: Cubic from Factored Form
BasicSketch .
End Behavior
Degree 3, positive leading coefficient → down-left, up-right (↘ ↗)
Zeros
— all with multiplicity 1
Multiplicity → All Cross (odd)
y-intercept
→ passes through the origin
Test Points & Sketch
| −3 | −1 | 1 | 4 | |
| −18 | 4 | −6 | 24 |
The graph enters from lower-left, crosses at , rises to a local max between and , crosses at origin, dips to a local min between and , crosses at , and exits upper-right.
Interactive Graph — verify your sketch:
Example 2: Degree 4 with a Bounce
IntermediateSketch .
End Behavior
Degree 4 (even), negative leading coefficient → both ends down (↘ ↘)
Zeros & Multiplicity
mult 2 → Bounce
mult 1 → Cross
mult 1 → Cross
y-intercept
Point:
Test Point
→ the graph is positive between and
The graph falls from upper-left, bounces at , dips down through , crosses at , rises to a peak near , crosses at , then falls to the lower-right.
Interactive Graph — verify your sketch:
Example 3: From Standard Form
AdvancedSketch .
End Behavior
Degree 4, positive leading coefficient → both ends up (↗ ↗)
Factor Completely
Zeros & Multiplicity
Test Point Between Zeros
→ the graph dips to a local min of 1 at
The graph rises from upper-left, bounces at the origin , rises to , then dips back down and bounces at , and rises again to upper-right. The graph never goes below the x-axis!
Interactive Graph — verify your sketch:
Common Pitfalls
Forgetting the y-intercept
The y-intercept is your anchor point. Forgetting it can lead to a sketch where the graph floats in the wrong region between zeros.
Drawing Sharp Corners
Polynomial graphs are always smooth, continuous curves — no sharp corners, no gaps, no vertical lines. If your sketch has a pointy angle, you need to smooth it out.
Real-Life Applications
Roller coaster designers use polynomial curves to model ride profiles. The zeros are where the track crosses a reference level, the turning points are the hills and valleys, and the end behavior determines how the ride starts and finishes. Your sketching skills translate directly to understanding these real-world engineering curves.
Practice Quiz
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