1. 9.3 – Rational Function and Their Graphs

2. Review: STEPS for GRAPHING
HOLES
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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

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

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

5. Finding holes and asymptotes
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
VA: x=-1, -5
HA: y=0 (power of the denominator is greater than the numerator)
Holes: none

6. Let’s try some Find the vertical, horizontal asymptotes and any holes
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

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

8. GRAPHING 1) HOLES? hole @ x = -1 VERTICAL ASYMPTOTES? None!
3) HORIZONTAL ASYMPTOTES?
None!
4) The graph -
Graph the function
y=x with a hole
at x=-1
What cancels? 8

9. GRAPHING 1) HOLES? hole @ x = 0 VERTICAL ASYMPTOTES?
Yes ! VA @ x =-2 , 5
3) HORIZONTAL ASYMPTOTES?
Yes ! HA @ y =0 (Power of the denominator is greater than the numerator)
4) T-CHART X 6
Y = 1/2
WAIT – What about the Horizontal Asymptote here? -3
Y = -5/8 1
Y = 1/12 2
Y = 0 3
Y = -1 / 10 9

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
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 -1
Y = 1/2 4
Y = -1/3
Right
Middle 2
Y = 0

11. Let’s try one: Sketch the Graph
1) HOLES?
none
VERTICAL ASYMPTOTES?
Yes ! VA @ x = 1
3) HORIZONTAL ASYMPTOTES?
Yes ! HA @ y =0 (Power of the denominator is greater than the numerator)
4) T-CHART X
Y = 0 -1
Y = 1/4 -2
Y = .22 2
Y=-2 3
Y = -3/4 11