Unit 7 –Rational Functions

Graphing Rational Functions

What to do first• FACTOR!!!!– Factor either numerator, denominator, or

both, before graphing.– Do NOT simplify/cancel anything… yet.

Graphing Rational Functions• To sketch these graphs, you must first

identify…

The

Mathtasitc 4!

M4: Vertical Asymptotes• Values of x that make the

denominator 0.• Ex: 𝑓 (𝑥 )= 4 𝑥

𝑥2−3𝑥−4After factoring we have:

𝑓 (𝑥 )= 4 𝑥(𝑥−4)(𝑥+1)

Denominator is 0 at x = 4 & x = -1. Those would be vertical asymptotes (graph cannot cross those lines).

M4: Zeros• Values of x that make

the numerator 0.• Ex: 𝑓 (𝑥 )=𝑥2+𝑥−6

𝑥−4After factoring we have:

𝑓 (𝑥 )=(𝑥+3)(𝑥−2)𝑥−4

Numerator is 0 at x = -3 & x = 2. Those points would be zeros (graph hits x-axis at those points).

M4: Holes• Values of x that make both

numerator & denominator 0.

• Ex:After factoring we have: 𝑓 (𝑥 )=(𝑥+2)(𝑥−2)

𝑥+2Numerator and denominator are 0 at x = -2. That point is a hole in the graph (graph passes through that point, but the function is undefined at that point).

𝑓 (𝑥 )=𝑥2−4𝑥+2

M4: Holes• Holes are NOT zeros.• They are not necessarily on the x-axis.– To find the coordinates of a hole, cancel the

common binomial, and plug the value of x into what’s left to find the y value.

After simplifying we have: 𝑓 (𝑥 )=𝑥−2Plugging -2 for x gives:

A hole would be located at the point (-2, -4).

M4: Horizontal Asymptote

• Determined by degrees of numerator and denominator.– If numerator degree > denominator degree,

no horizontal asymptote.– Ex. 𝑓 (𝑥 )=𝑥2+𝑥−6

𝑥−4Numerator degree = 2, denominator degree = 1. No horizontal asymptote.

M4: Horizontal Asymptote

• Determined by degrees of numerator and denominator.– If numerator degree < denominator degree,

there is a horizontal asymptote at y = 0.– Ex.

𝑓 (𝑥 )= 4 𝑥𝑥2−3𝑥−4

Numerator degree = 1, denominator degree = 2. Horizontal asymptote at y = 0.

M4: Horizontal Asymptote

• Determined by degrees of numerator and denominator.– If numerator degree = denominator degree,

the horizontal asymptote is at y = ratio of leading coefficients.

– Ex. 𝑓 (𝑥 )=3 𝑥2−5𝑥−8

𝑥2−3 𝑥−4Degrees are both 2. Ratio of leading coefficients = 3/1. Horizontal asymptote at y = 3.

Identifying the Mathtastic 4• After finding

asymptotes, zeros, and holes, graphs of rational functions are easy to sketch.– Be sure to use your

graphing calculator to check your work.

Identifying the Mathtastic 4• Practice identifying the

Mathtastic 4 with the functions presented in this presentation.– Keep in mind that all 4 will

not always show up in a single function.

𝑓 (𝑥 )= 4 𝑥𝑥2−3𝑥−4

𝑓 (𝑥 )=𝑥2+𝑥−6𝑥−4

𝑓 (𝑥 )=3 𝑥2−5𝑥−8

𝑥2−3 𝑥−4 𝑓 (𝑥 )=𝑥2−4𝑥+2

Homework

Textbook Section 8-4 (pg. 598): 33-42Should be completed before Unit 7 Exam