Math-3 Lesson 3-6 Analyze Rational functions The Oblique Asymptote
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1 Math- Lesson - Analyze Rational functions The Oblique Asymptote
2 Quiz: a What is the domain? b Where are the holes? c What is the vertical asymptote? y a -, b = c = -
3 Last time Zeroes of the numerator are -intercepts (or are imaginary Zeroes of the denominator are either ( Vertical asymptotes or ( Holes The zeroes of the denominator that simplify (and disappear causes holes in the graph.
4 Your turn: What are the equations of the horizontal and vertical asymptotes? f ( horizontal asymptote is: y = Vertical asymptote is: =
5 Your turn: What are the equations of: a Horizontal asymptote? Range =? b Vertical asymptote? Domain =? 7 a y = 7, range: y 7 b =, domain: f 7 ( ( 5 a y = 5, range: y 5 b = -, domain: - h( ( 4 a y = -4, range: y -4 b =, domain: -
6 f ( Domain : 4 4 = - is a zero of the numerator: -intercept at = -. = 4 is a zero of the denominator: vertical asymptote at = 4. f( =? gives the y-intercept: y 4 (, 4
7 Your Turn: (a What is the domain of the function? And Domain:, - Specify the value of that will be: (b a hole or (c a vertical asymptote 5 ( ( ( y ( ( ( 4 hole at = - Vertical asymptote at =
8 Your Turn: (a What is the domain of the function? And Domain: 4, - Specify the value of that will be: (b a hole or (c a vertical asymptote 8 4 ( ( ( y ( ( 4 ( 4 8 hole at = - Vertical asymptote at = 4
9 Vertical asymptote: the value of that makes the denominator = zero. -intercept: the value of that makes the numerator = zero. Only the numerator makes a fraction equal to zero!
10 Horizontal Asymptote: think end behavior. ( Use long division. X y X y? Let s think about this!!!! What value does a fraction approach if the denominator gets bigger and bigger (forever?,, The fraction approaches zero. Horizontal asymptote: y =
11 Horizontal asymptote: there are different methods to find this. Not all rational functions have a horizontal asymptote. ( factor out the largest power of in the denominator. Then figure out the right end behavior. X y? / is the reciprocal function. X / ( ( As gets bigger and bigger what does become?
12 Your turn: What are the -intercepts and vertical asymptotes? Vertical asymptote: the value of that makes the denominator = zero. -intercept: the value of that makes the numerator = zero.
13 What is the horizontal asymptote? (quotient of long division Horizontal Asymptote: y = if degree of the denominator > degree of numerator.
14 Horizontal Asymptote: Factor out the largest power of in the denominator then find the horizontal asymptote using end behavior. X g ( X y? ( X HA: y =
15 Your turn: Find horizontal asymptote: (quotient of long division 7 ( g Horizontal asymptote: y = ( g X
16 Your turn: Find the domain, -intercepts, holes, vertical asymptotes, and the horizontal asymptote: (HINT: simplify first.. 7 ( ( ( ( 7 Domain: ( ( 7 -intercept: = hole: = - Vertical asymptote: = 7 Horizontal asymptote ( ( 7 ( ( 7 X -/ -7/ Horizontal asymptote: y =
17 7 ( ( 7 Domain: hole: = - -intercept: = Vertical asymptote: = 7 Horizontal asymptote: y =
18 Oblique Asymptote (quotient of long division Numerator degree > Denominator degree Oblique asymptote: =
19 Oblique Asymptote: Numerator Degree > denominator degree ( X (goes up There is no Horizontal Asymptote This method of determining end behavior only distinguishes between number (horizontal asymptote +/- (goes up or down
20 Rules to Predict End Behavior Using the relative degree between the numerator and the denominator.. If: degree numerator (DN < degree denominator (DD then: +/-, y. If: degree numerator (DN = degree denominator (DD then: +/-, y ratio of the lead coefficients. If: degree numerator (DN > degree denominator (DD then: +/-, y must use either synthetic or long division to rewrite the rational equation with end term being remainder/divisor which disappears as
21 Horizontal asymptote: think end behavior. ( Use long division. X y X y? Let s think about this!!!! What value does a fraction approach if the denominator gets bigger and bigger (forever?,, The fraction approaches zero. Horizontal asymptote: y =
22 Your turn: use long division to find the horizontal asymptote. ( 5 5 X y? 5 X The fraction approaches zero. Horizontal asymptote: y =
23 Will there be a horizontal asymptote? Use long division. 4 ( g 4 ( 7 7 ( g X y? The fraction approaches zero. 7 ( g X + Oblique asymptote: y = + 4 (
24 Your turn: find the oblique asymptote Use long division. 5 ( g 5 4 ( 4 4 ( g X y? The fraction approaches zero. 4 ( g X + Oblique asymptote: y = (
25 ( g Your turn: find the asymptote Use long division. 4 ( g 4 ( ( g X y? The fraction approaches zero. X Oblique asymptote: 4 ( y
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