start up day 14 write a polynomial function of minimum degree with integer coefficients given the...
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Start Up Day 14WRITE A POLYNOMIAL FUNCTION OF MINIMUM DEGREE WITH
INTEGER COEFFICIENTS GIVEN THE FOLLOWING ZEROS:
Questions?
OBJECTIVE: Students will be able to determine, analyze and sketch the graph of a rational function.
ESSENTIAL QUESTION: How do you determine the domain of a rational function? What types of asymptotes are possible? How can you determine if a rational function has a slant asymptote or an end behavior polynomial asymptote?
CLASS-WORK/HOME LEARNING: Pg. 225# 4, 7, 11‐ 14, 24, 39, 45, 57 /Pg. 225 # 2, 10, 21, 29, 43, 46, 61, 65‐68
OCTOBER 1, 2015
Graphs of Rational Functions
Domain of Rational Functions
Rational functions are fractions. Therefore, how would you find restrictions on your domain?
1) 2)
5 2( )
3 7
xf x
x
2
11 6( )
10 24
xg x
x x
Denominator cannot be equal to zero.
Transformations of Rational Functions
Translations, dilations, and reflections of which basic
function?
3) 4)
1( )
4h x
x
2 5
( )3
xp x
x
X-intercepts of Rational Functions
Set the numerator equal to zero and solve.
5) 6)2 19 42
( )8 3
x xm x
x
22 9 5( )
12 7
x xn x
x
Y-intercepts of Rational Functions
The ratio of the numerator’s constant and the denominator’s constant or f(0).
7) 8)2 9 20
( )7 4
x xr x
x
2 5 24( )
11 6
x xs x
x
Vertical Asymptotes of Rational Functions
Set the denominator equal to zero and solve.
9) 10)2
3 5( )
31 66
xt x
x x
2
9 5( )
3 19 14
xw x
x x
Horizontal Asymptotes of Rational Functions
3 cases… If the numerator’s degree is lower than the denominator’s
degree, the horizontal asymptote is y = 0. If the numerator’s degree is equal to the denominator’s degree,
the horizontal asymptote is y = ratio of the leading coefficients. If the numerator’s degree is greater than the denominator’s
degree, there is no horizontal asymptote.
11) 12) 13)2
2 5( )
4 11
xb x
x x
2
2
6 7 12( )
3 14 8
x xc x
x x
23 11( )
8 5
xd x
x
Slant Asymptotes of Rational Functions
22 5 11( )
3
x xk x
x
***If there is no horizontal asymptote, then check for a slant asymptote. Specifically, if there is a difference of one degree, we can find a slant asymptote. ***
14) 15)
24 7 3( )
5 2
x xf x
x
End Behavior of Rational Functions
End behavior in rational functions asymptotes is defined by the quotient calculated through long division. These end behavior asymptotes can be constant (horizontal), slant (linear) or polynomial.
16) 17) 18) 2
2
4 5 11( )
3 12 1
x xk x
x x
25 8 13( )
4
x xt x
x
Summary: