1. Holes & Slant Asymptotes
Graphing Rationals
Holes & Slant Asymptotes

2. RECAP  Characteristics of Rational Functions that you need to know!!!
Domain/Range
Zeros/Roots/Solutions/x-Intercepts
y-Intercept
All Asymptotes (Vertical/Horizontal/Slant)
Points of Discontinuity (POD for short)
Intervals of Increase/Decrease
Relative & Absolute Extrema (Min/Max)
End Behavior
**Average Rate of Change

3. Quick Review: Identify the Characteristics
Domain:__________
Range:__________
Zeros:__________
y-Intercept:__________
POD:__________
Inc/Dec:__________
Extrema:__________
End Behavior:__________
AROC:
[-5,0] __________
𝒇 𝒙 = 𝒙+𝟓 𝟐𝒙−𝟑

4. Quick Review: Identify the Characteristics
Domain:__________
Range:__________
Zeros:__________
y-Intercept:__________
POD:__________
Inc/Dec:__________
Extrema:__________
End Behavior:__________
AROC:
[-2,2] __________
𝒇 𝒙 = 𝒙+𝟐 𝒙 𝟐 −𝒙−𝟏𝟐

5. HOLES in Rational Functions
We started the unit on Rational Functions by simplifying Rational expressions… This is important because before we do any graphing, we always need to look at the simplified version of the Rational Function. If any common algebraic factors cancel out in the numerator and denominator, then the graph looks like the resulting Rational Function with a HOLE at the canceled out x-value! EX 𝒙 𝟐 +𝟓𝒙+𝟔 𝒙+𝟐 = (𝒙+𝟐)(𝒙+𝟑) 𝒙+𝟐 =𝒙+𝟑
**This function looks like the linear function x + 3, but there is a HOLE in it at x = -2

6. What does that look like??
𝒇 𝒙 = 𝒙 𝟐 +𝟓𝒙+𝟔 𝒙+𝟐

7. RECAP: Horizontal Asymptotes
Remember that the degree is the highest exponent.
Remember that the leading coefficient is the number in front of your “degree” variable.
Given a rational function 𝑵(𝒙) 𝑫(𝒙) , compare the degrees of the
numerator  𝑵 𝒙 and the denominator  𝑫(𝒙).
There are 3 Possibilities:
Deg. of N(x) < Deg. of D(x); asymptote at y = 0
Deg. of N(x) = Deg. of D(x); asymptote at
Deg. of N(x) > Deg. of D(x); NO horizontal asymptote!
**Possible slant asymptote here…more to come on that.
Leading Coefficient of Numerator
Leading Coefficient of Denominator

8. Slant Asymptotes
**When the degree on top is exactly one more than the degree on bottom, we will have a slant asymptote. To find the SLANT ASYMPTOTE (it will just be a line y = mx + b), use synthetic or long division… EX  𝒙 𝟐 +𝟓𝒙+𝟔 𝒙−𝟏 Using polynomial division, we get 𝒙+𝟔+ 𝟏𝟐 𝒙−𝟏 We disregard the remainder, and the resulting linear binomial is our SLANT ASYMPTOTE!!

9. What does that look like??
𝒇 𝒙 = 𝒙 𝟐 +𝟓𝒙+𝟔 𝒙−𝟏

10. Identify the Characteristics
Domain:__________
Range:__________
Zeros:__________
y-Intercept:__________
POD:__________
Inc/Dec:__________
Extrema:__________
End Behavior:__________
AROC:
[0,4] __________
𝒇 𝒙 = 𝒙 𝟐 +𝟑𝒙−𝟒 𝒙+𝟒

11. Identify the Characteristics
Domain:__________
Range:__________
Zeros:__________
y-Intercept:__________
POD:__________
Inc/Dec:__________
Extrema:__________
End Behavior:__________
AROC:
[0,4] __________
𝒇 𝒙 = 𝒙 𝟐 +𝟓𝒙+𝟔 𝒙+𝟏