Holes & Slant Asymptotes Graphing Rationals Holes & Slant Asymptotes

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

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

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

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

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

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

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!!

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

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

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