unit 7 –rational functions
DESCRIPTION
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!. - PowerPoint PPT PresentationTRANSCRIPT
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Unit 7 –Rational Functions
Graphing Rational Functions
![Page 2: Unit 7 –Rational Functions](https://reader035.vdocument.in/reader035/viewer/2022062410/5681610f550346895dd0674d/html5/thumbnails/2.jpg)
What to do first• FACTOR!!!!– Factor either numerator, denominator, or
both, before graphing.– Do NOT simplify/cancel anything… yet.
![Page 3: Unit 7 –Rational Functions](https://reader035.vdocument.in/reader035/viewer/2022062410/5681610f550346895dd0674d/html5/thumbnails/3.jpg)
Graphing Rational Functions• To sketch these graphs, you must first
identify…
The
Mathtasitc 4!
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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).
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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).
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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
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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).
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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.
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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.
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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.
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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.
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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
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Homework
Textbook Section 8-4 (pg. 598): 33-42Should be completed before Unit 7 Exam