9.3 – Rational Function and Their Graphs. Review: STEPS for GRAPHING HOLES ___ EX _.

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Presentation transcript:

9.3 – Rational Function and Their Graphs

Review: STEPS for GRAPHING HOLES ___________________________________________ EX _________________________________________ Discontinuous part of the graph where the line jumps over. Represented by a little open circle. x = 3 x = 2 No hole at x = 0

VERTICAL ASYMPTOTES ___________________________________________ EX _________________________________________ Discontinuous part of the graph where the line cannot cross over. Represented by a dotted line called an asymptote. x = 2 x =0 x = 2, -5 Review: STEPS for GRAPHING

HORIZONTAL ASYMPTOTES n = degree of numerator d = degree of denominator _______________________________________________ Case 1 n > d No HA Case 2 n < d y = 0 Case 1 n = d HA is the ratio of coefficients y = 4 / 5 Review: STEPS for GRAPHING

Finding holes and asymptotes VA: x=-1, -5 HA: y=0 (power of the denominator is greater than the numerator) Holes: none VA: none (graph is the same as y=x-1 once the (x-2)s cancel HA: none (degree of the numerator is greater than the denominator) Hole: x=2

Let’s try some VA: x=3 HA: none (power of the numerator is greater than the denominator) Holes: x=2 VA: x=-5,0 ( cancel the (x-3)s HA: y=0 (degree of the denominator is greater than the numerator) Hole: x=3 Find the vertical, horizontal asymptotes and any holes

GRAPHING y = x / (x – 3) 1) HOLES? no holes since nothing cancels 2) VERTICAL ASYMPTOTES? Yes ! x =3 4) T-CHART X Y = x/(x – 3) 4Y = 4 2Y = -2 3) HORIZONTAL ASYMPTOTES? Yes ! y =1 0 5 Y = 0 Y = 5/2

GRAPHING 1) HOLES? 2) VERTICAL ASYMPTOTES? 3) HORIZONTAL ASYMPTOTES? 4) The graph - What cancels? Graph the function y=x with a hole at x=-1 x = -1 None!

GRAPHING 1) HOLES? 2) VERTICAL ASYMPTOTES? 4) T-CHART X 6Y = 1/2 -3Y = -5/8 3) HORIZONTAL ASYMPTOTES? 1 2 Y = 1/12 Y = 0 3 Y = -1 / 10 WAIT – What about the Horizontal Asymptote here? x = 0 Yes ! x =-2, 5 Yes ! y =0 (Power of the denominator is greater than the numerator)

Remember, Horizontal Asymptotes only describe the ends of the function (left and right). What happens in the middle is ‘fair game’. T-CHART X Y = 1/2 4Y = -1/3 2 Y = 0 To find out what the graph looks like between the vertical asymptotes, go to a T Chart and plug in values close to the asymptotes. Left Right Middle

Let’s try one: Sketch the Graph 1) HOLES? 2) VERTICAL ASYMPTOTES? 4) T-CHART X 0 Y = 0 Y = 1/4 3) HORIZONTAL ASYMPTOTES? -2 2 Y =.22 Y=-2 3 Y = -3/4 none Yes ! x = 1 Yes ! y =0 (Power of the denominator is greater than the numerator)