8.1/8.2- Graphing Rational Functions WILL TAKE THE FULL CLASS!!!

Definition Parent reciprocal function: 𝑓 𝑥 = 1 𝑥 , where 𝑥≠0 x y 1 2 10 1/2 -1 -2 -10 -1/2

Definition (cont.) Vertical asymptote: Imaginary vertical line which you can NEVER cross Horizontal asymptote: Imaginary horizontal line which you can POSSIBLY cross Parent reciprocal function VA: 𝑥=0 HA: 𝑦=0

Transformations VA: 𝒙=𝒉 HA: 𝒚=𝒌 𝑦= 𝑎 𝑥−ℎ +𝑘 𝑎: Reflection or vertical stretch or shrink 𝑎<0: reflection over x-axis 𝑎>1: vertical stretch 0<𝑎<1: vertical compression ℎ: Horizontal translation Minus in the middle means go right Plus in the middle means go left 𝑘: Vertical translation Plus means go up Minus means go down VA: 𝒙=𝒉 HA: 𝒚=𝒌

Ex: For 𝒚= 𝟖 𝒙 , find: 𝑎= ℎ= 𝑘= Transformation: VA: HA: Graph. X-int: Y-int: Domain: Range:

On Your Own: For 𝒚= 𝟑 𝒙 , find: 𝑎= ℎ= 𝑘= Transformation: VA: HA: Graph. X-int: Y-int: Domain: Range:

Ex: For 𝒚= −𝟐 𝒙 , find: 𝑎= ℎ= 𝑘= Transformation: VA: HA: Graph. X-int: Y-int: Domain: Range:

Ex: For 𝒚= 𝟏 𝒙+𝟏 −𝟐, find: 𝑎= ℎ= 𝑘= Transformation: VA: HA: Graph. X-int: Y-int: Domain: Range:

On Your Own: For 𝒚= 𝟏 𝒙−𝟒 +𝟔, find: 𝑎= ℎ= 𝑘= Transformation: VA: HA: Graph. X-int: Y-int: Domain: Range:

Answer: (A) because it moved 3 left (h = -3) and 4 up (k = 4) Ex: Answer: (A) because it moved 3 left (h = -3) and 4 up (k = 4)

On Your Own: This graph of a function is a translation of the graph of 𝑦= 2 𝑥 . What is an equation for the function? Answer: 𝑦= 2 𝑥−1 −4 because it moved 1 right (h = 1) and 4 down (k = -4)

Def Rational functions: Fractions of polynomial functions When is a fraction undefined? Domain of a rational function = all values except when you cancel a factor or when the denominator is 0. If you have a denominator left over, the value that makes that denominator 0 makes a vertical asymptote

Continuous vs. Discontinuous Continuous: When you cannot cancel a factor and the denominator is never 0 This means when you draw the graph, your pencil never leaves the paper Ex: 1st graph Discontinuous: When you cancel a factor or the denominator can be 0 This means when you draw the graph, your pencil must leave the paper Ex: 2nd and 3rd graph

Discontinuities If you cancel a factor, you create a hole at that spot Ex: The graph of 𝑦= (𝑥+3)(𝑥+2) 𝑥+2 has a hole at 𝑥=−2 Holes are also called removable discontinuities If you still have a denominator left over, you have a vertical asymptote at that spot Ex: The graph of 𝑦= 𝑥+4 𝑥−2 has a vertical asymptote at 𝑥=2 START FROM HERE NEXT TIME

Domain= all real numbers except 𝑥=3 and 𝑥=1 (VA’s) Find the domain of each rational function. Identify what occurs at those points. Find the x & y-intercepts. a) 𝑦= 𝑥+3 𝑥 2 −4𝑥+3 Answers: Domain= all real numbers except 𝑥=3 and 𝑥=1 (VA’s) x-int = (-3 , 0) y-int = (0 , 1)

Domain= all real numbers except 𝑥=4 (hole) Find the domain of each rational function. Identify what occurs at those points. Find the x & y-intercepts. b) 𝑦= 𝑥 2 −3𝑥−4 𝑥−4 Answers: Domain= all real numbers except 𝑥=4 (hole) x-int = (-1 , 0) y-int = (0 , 1)

Domain= all real numbers Ex: Find the domain of each rational function. Identify what occurs at those points. Find the x & y-intercepts. c) 𝑦= 𝑥−5 𝑥 2 +1 Answers: Domain= all real numbers x-int = (5 , 0) y-int = (0 , -5)

Horizontal Asymptotes To find a HA, compare the degree of the numerator and denominator If the degree of the numerator is m and the degree of the denominator is n, then either of these can happen: For 𝑚<𝑛, HA at 𝑦=0 (the x-axis) For 𝑚=𝑛, HA at 𝑦= 𝑎 𝑏 where a is the coefficient of the leading term in the numerator and b is the coefficient of the leading term in the denominator For 𝑚>𝑛, no HA

Ex: What is the horizontal asymptote for: 𝑦= 2𝑥 𝑥−3 b) 𝑦= 𝑥−2 𝑥 2 −2𝑥−3 c) 𝑦= 𝑥 2 2𝑥−5 Answers: 𝑦=2 𝑦=0 No HA

Ex: For 𝑦= 𝑥+4 𝑥−4 , Hole = ________ VA: __________ HA: __________ X-int = _______ Y-int = _______ Graph the rational function

Ex: For 𝑦= 𝑥+6 𝑥 2 +𝑥−6 , Hole = ________ VA: __________ HA: __________ X-int = _______ Y-int = _______ Graph the rational function

Ex: For 𝑦= 𝑥 2 −4 3𝑥−6 , Hole = ________ VA: __________ HA: __________ X-int = _______ Y-int = _______ Graph the rational function

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