9.3 – rational function and their graphs. review: steps for graphing holes...
DESCRIPTION
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 GRAPHINGTRANSCRIPT
9.3 – Rational Function and Their Graphs
Review: STEPS for GRAPHING
HOLES
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EX _________________________________________
EX _________________________________________
Discontinuous part of the graph where the line jumps over.
Represented by a little open circle.
)5x)(3x()3x(y
)2x(x)2x(xy 2
Hole @ x = 3
Hole @ x = 2No hole at x = 0
VERTICAL ASYMPTOTES
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EX _________________________________________
EX _________________________________________
Discontinuous part of the graph where the line cannot cross over.
Represented by a dotted line called an asymptote.
)2x()5x(y
)5x)(2x(xxy
VA @ x = 2
Hole @ x =0VA @ x = 2, -5
Review: STEPS for GRAPHING
HORIZONTAL ASYMPTOTESn = degree of numeratord = degree of denominator
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Case 1 n > d )2x(7x5y
2
No HA
Case 2 n < d 1x3xy 3
HA @ y = 0
Case 1 n = d )2x)(2x(51x4y
2
HA is the ratio of coefficientsHA @ 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 ! VA @ x =3
4) T-CHART
X Y = x/(x – 3)
4 Y = 4
2 Y = -2
3) HORIZONTAL ASYMPTOTES? Yes ! HA @ 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 functiony=x with a holeat x=-1
hole @ x = -1
None!
None!
GRAPHING
1) HOLES?
2) VERTICAL ASYMPTOTES?
4) T-CHART
X
6 Y = 1/2
-3 Y = -5/8
3) HORIZONTAL ASYMPTOTES?
1
2
Y = 1/12
Y = 0
)5x)(2x(x)2x(xy
)5x)(2x()2x(y
)5x)(2x()2x(y
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)
Remember, Horizontal Asymptotes only describe the ends of the function (left and right). What happens in the middle is ‘fair game’.
T-CHART
X
-1 Y = 1/2
4 Y = -1/3
2 Y = 0
)5x)(2x()2x(y
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
-1 Y = 1/4
3) HORIZONTAL ASYMPTOTES?
-2
2
Y = .22
Y=-23 Y = -3/4
none
Yes ! VA @ x = 1
Yes ! HA @ y =0 (Power of the denominator is greater than the numerator)