9.3 Graphing General Rational Functions

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9.3 Graphing General Rational Functions

• p. 547

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Steps to graphing rational functions

1. Find the y-intercept.
2. Find the x-intercepts.
3. Find vertical asymptote(s).
4. Find horizontal asymptote.
5. Find any holes in the function.
6. Make a T-chart choose x-values on either side
   between all vertical asymptotes.
7. Graph asymptotes, pts., and connect with curves.
8. Check your graph with the calculator.

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How to find the intercepts

• y-intercept
• Set the y-value equal to zero and solve
• x-intercept
• Set the x-value equal to zero and solve

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How to find the vertical asymptotes

• A vertical asymptote is vertical line that the
  graph can not pass through. Therefore, it is the
  value of x that the graph can not equal. The
  vertical asymptote is the restriction of the
  denominator!
• Set the denominator equal to zero and solve

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How to find the Horizontal Asymptotes

• If degree of top lt degree of bottom, y0
• If degrees are ,
• If degree of top gt degree of bottom, no horiz.
  asymp, but there will be a slant asymptote.

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How to find slant asymptotes

• Do synthetic division (if possible) if not, do
  long division!
• The resulting polynomial (ignoring the remainder)
  is the equation of the slant asymptote.

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How to find the points of discontinuity (holes)

• When simplifying the function, if you cancel a
  polynomial from the numerator and denominator,
  then you have a hole!
• Set the cancelled factor equal to zero and solve.

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Steps to graphing rational functions

1. Find the y-intercept.
2. Find the x-intercepts.
3. Find vertical asymptote(s).
4. Find horizontal asymptote.
5. Find any holes in the function.
6. Make a T-chart choose x-values on either side
   between all vertical asymptotes.
7. Graph asymptotes, pts., and connect with curves.
8. Check your graph with the calculator.

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Ex Graph. State domain range.
5. Function doesnt simplify so NO HOLES!

• 2. x-intercepts x0
• 3. vert. asymp. x210
• x2 -1
• No vert asymp
• 4. horiz. asymp
• 1lt2 (deg. top lt deg. bottom)
• y0

6. x y -2 -.4 -1 -.5
0 0 1 .5 2 .4
(No real solns.)
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Domain all real numbers Range
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Ex Graph then state the
domain and range.
6. x y 4 4 3 5.4 1
-1 0 0 -1 -1 -3
5.4 -4 4

• 2. x-intercepts
• 3x20
• x20
• x0
• 3. Vert asymp
• x2-40
• x24
• x2 x-2
• 4. Horiz asymp
• (degrees are )
• y3/1 or y3

On right of x2 asymp.
Between the 2 asymp.
On left of x-2 asymp.
5. Nothing cancels so NO HOLES!
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Domain all real s except -2 2 Range all
real s except 0ltylt3
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Ex Graph, then state the domain range.

• y-intercept -2
• x-intercepts
• x2-3x-40
• (x-4)(x1)0
• x-40 x10
• x4 x-1
• Vert asymp
• x-20
• x2
• Horiz asymp 2gt1
• (deg. of top gt deg. of bottom)
• no horizontal asymptotes, but there is a slant!

5. Nothing cancels so no holes.
6. x y -1 0 0
2 1 6 3 -4 4 0
Left of x2 asymp.
Right of x2 asymp.
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Slant asymptotes

• Do synthetic division (if possible) if not, do
  long division!
• The resulting polynomial (ignoring the remainder)
  is the equation of the slant asymptote.
• In our example
• 2 1 -3 -4
• 1 -1 -6

Ignore the remainder, use what is left for the
equation of the slant asymptote yx-1
2 -2
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Domain all real s except 2 Range all real
s
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Assignment Workbook page 611-9Find each
piece of the function.