Today in Pre-Calculus
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Transcript of Today in Pre-Calculus
Today in Pre-Calculus• No calculators needed• Notes:
– Rational Functions and Equations– Transformations of the reciprocal
function
• Go over quiz• Homework
Rational Functions
A rational function, f(x), is a ratio or quotient of polynomial functions p(x) and q(x) expressed as
* The domain of f(x) is all real numbers except where q(x) = 0
( )( )
( )
p xf x
q x
Rational FunctionsAre the following rational functions? If yes, state the domain.
2
3
2
2
3
1 5. ( ) . ( )
1
3 5. ( )
4
4 7 16 64. ( ) . ( )
10 5 3 1
xa f x d i x
x x
x xb g x
x
x xc j x e h x
x x x
yes. D:(-∞,0) υ(0,∞)
yes. D:(-∞,-2) υ (-2,2) υ (2,∞)
No, numerator not a polynomial
yes. D: (-∞,∞)
:
: , 2 2,0 0, 2 2,
yes
D
Transformations of the Reciprocal Function
The simplest rational function is the basic function, 1
( )f xx
Horizontal asymptote:y=0Vertical asymptote:x=0
Example 1Sketch the graph and find an equation for the function g whose graph is obtained from the reciprocal function, by a translation of 2 units to the right.
1( )f x
x
1( )
2g x
x
Example 2Sketch the graph and find an equation for the function g whose graph is obtained from the reciprocal function, by a translation of 5 units to the right, followed by a reflection across the x-axis
1( )f x
x
1( )
5g x
x
Example 3Sketch the graph and find an equation for the function g whose graph is obtained from the reciprocal function, by a translation of 4 units to the left, followed by a vertical stretch by a factor of 3, and finally a translation 2 units down.
1( )f x
x
3( ) 2
4g x
x
Graphing Rational FunctionsThe graph of any rational function of the form can be obtained by transforming by using polynomial long division.
1( )f x
x
( )ax b
f xcx d