Graphing Rational Functions: Key Features

Lesson 1 of 2

In this lesson:

• Zeros, holes, vertical asymptotes
• Horizontal asymptotes from degree comparison

What You Will Learn Today

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

1. Find zeros of a rational function from the numerator
2. Identify holes from shared factors in numerator and denominator
3. Find vertical asymptotes from uncanceled denominator zeros
4. Determine horizontal asymptotes by comparing degrees

Rational Functions Extend Polynomial Graphs

You know polynomial graphs — zeros and end behavior.

Rational functions add new behaviors:

• Vertical asymptotes — the function blows up at certain x-values
• Holes — single missing points in an otherwise continuous graph
• Horizontal asymptotes — the function levels off instead of growing without bound

Zeros Come from the Numerator

For , zeros occur where , provided

Step 1: Factor numerator and denominator completely

Step 2: Find zeros of numerator

Step 3: Check each numerator zero against denominator

Example: Zeros with Denominator Check

• Numerator zeros: and
• Check denominator: ;
• Both are valid zeros → x-intercepts at and

Shared Factors Create Holes, Not Zeros

When the same factor appears in both numerator and denominator, it creates a hole — a removable discontinuity.

The factor cancels, but is still excluded from the domain.

Locating a Hole Using the Simplified Form

For : the factor cancels

• excluded from domain → hole at
• Substitute into simplified form:
• Hole at — mark with an open circle

Check-In: Hole or Vertical Asymptote?

At : does the graph have a hole or a vertical asymptote?

Factor first, then check whether the problematic factor cancels.

Check-In Answer: That Is a Hole

• : factor cancels → hole at
• : factor does not cancel → vertical asymptote

Rule: cancel = hole; no cancel = vertical asymptote

Vertical Asymptotes: Blow-Up at the Boundary

Vertical asymptote at : denominator zero that does not cancel

Near , or — the function "blows up"

The graph approaches the dashed vertical line but never touches it

The Function Explodes Near a VA

VA at . As , .

Two Vertical Asymptotes Create Three Regions

• Zero at (denominator: ✓)
• VAs at and
• Three separate graph regions divided by the two VAs

Find the Vertical Asymptotes: Your Turn

Which values of give vertical asymptotes?

is a numerator zero — an x-intercept, not a VA.

Checking Your Vertical Asymptote Answer

• VAs at and (uncanceled denominator zeros)
• is an x-intercept — zero of numerator only

Horizontal Asymptotes from Degree Comparison

Horizontal asymptote: y-value approaches as

Compare degree of numerator () vs. denominator ():

• : HA at
• : HA at
• : no HA

HA When Numerator Degree Is Smaller

• Denominator grows faster → fraction approaches 0
• HA at (the x-axis)

HA Case 2: Equal Degrees

• Leading coefficients: num = 2, den = 3
• HA at
• Verify: ,

HA When Numerator Degree Is Larger

• Numerator grows faster → function grows without bound
• No horizontal asymptote

HA Check-In: Apply the Rule

What is the horizontal asymptote?

Step 1: Compare degrees. Step 2: If equal, take the ratio of leading coefficients.

Checking Your Horizontal Asymptote Answer

• Degrees equal (both 2) → ratio of leading coefficients
• HA at

Four Key Features of Rational Functions

Coming Up: Sign Analysis and Sketching

Lesson 2 of 2:

• Seven-step complete sketching procedure
• Sign analysis: which way does the curve approach each asymptote?
• Technology verification of hand sketches
• A function CAN cross a horizontal asymptote

Have your zeros, VAs, and HA ready — you'll use them all.