1. 9.3 – Rational Function and Their Graphs

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

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

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

5. 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

6. 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

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

8. 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 hole @ x = -1 None!

9. 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? hole @ x = 0 Yes ! VA @ x =-2, 5 Yes ! HA @ y =0 (Power of the denominator is greater than the numerator)

10. 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

11. 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 ! VA @ x = 1 Yes ! HA @ y =0 (Power of the denominator is greater than the numerator)