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Factor the following completely:Factor the following completely:
1. 3x2-8x+4
2. 11x2-99
3. 16x3+128
4. x3+2x2-4x-8
5. 2x2-x-15
6. 10x3-80
(3x-2)(x-2)(3x-2)(x-2)
11(x+3)(x-3)11(x+3)(x-3)
16(x+2)(x16(x+2)(x22-2x+4)-2x+4)
(x-2)(x+2)(x-2)(x+2)22
(2x+5)(x-3)(2x+5)(x-3)
10(x-2)(x10(x-2)(x22+2x+4)+2x+4)
9.3 9.3 Graphing General Rational Graphing General Rational
FunctionsFunctions
By: L. Keali’i AliceaBy: L. Keali’i Alicea
In the past, we graphed rational In the past, we graphed rational functions where x was to the first functions where x was to the first
power only. power only. What if x is not to the first power?What if x is not to the first power?
Such as: Such as:
1)(
2 x
xxf
Steps to graph when x is not to the 1st power
1. Find the x-intercepts. (Set numer. =0 and solve)2. Find vertical asymptote(s). (set denom=0 and
solve)3. Find horizontal asymptote. 3 cases:
a.a. If degree of top < degree of bottom, y=0If degree of top < degree of bottom, y=0b. If degrees are =, c. If degree of top > degree of bottom, no horiz.
asymp, but there will be a slant asymptote.4. Make a T-chart: choose x-values on either side &
between all vertical asymptotes.
5.Graph asymptotes, pts., and connect with curves.
6.Check solutions on calculator.
bottom of coeff. lead.
topof coeff. lead.y
Ex: Graph. State domain & range.Ex: Graph. State domain & range.
Slant asymptotesSlant asymptotes• Do synthetic division (if possible); if not, do long Do synthetic division (if possible); if not, do long
division!division!
• The resulting polynomial (ignoring the The resulting polynomial (ignoring the remainder) is the equation of the slant remainder) is the equation of the slant asymptote.asymptote.
In our example:In our example:
2 1 -3 -42 1 -3 -4
1 -1 -61 -1 -6
2 -22 -2
Ignore the remainder, Ignore the remainder, use what is left for the use what is left for the equation of the slant equation of the slant asymptote: y=x-1asymptote: y=x-1