Goals
Often in calculus we analyze functions by breaking them down into
smaller parts. This lesson presents this both in graphing and
algebra.
We will
investigate transformations of graphs
use arithmetic to analyze functions
review or learn an algorithm for dividing polynomials
Experiment
1
Use some technology to graph each of the following functions.
2
For each row identify a pattern (e.g., what does a negative in
front do to a graph).
3
Record these results to discuss in class.
f (x ) = x 2 + 2x + 1
f (x ) + 5
f (x + 2)
−f (x )
f (−x )
2f (x )
f (2x )
g(x ) = x 3 − x 2
g(x ) + 5
g(x + 2)
−g(x )
g(−x )
2g(x )
g(2x )
h(x ) = x 3 − x
h(x ) + 5
h(x + 2)
−h(x )
h(−x )
2h(x )
h(2x )
k(x ) = x 2
k(x ) + 5
k(x + 2)
−k(x )
k(−x )
2k(x )
k(2x )
Function Arithmetic
Just as numbers can be added, subtracted, multiplied, and divided,
polynomials can as well. For example
(x 2 + 2)(5x − 3) + 7
can be treated as
f (x )g(x ) + h(x )
where f (x ) = x 2 + 2, g(x ) = 5x − 3, and h(x ) = 7.
Function Arithmetic
As well as adding, subtracting, multiplying, and dividing,
polynomials can be composed as well. For example
(x 2 + 2)5 + 7
can be treated as
f (g(x ))
where f (x ) = x 5 + 7 and g(x ) = x 2 + 2.
Expand
f (x ) = x 4 + 3x 2 .
g(x ) = 2x 3 − 7x .
Expand the following.
1
f (2x )
2
f (−x )
3
g(−x )
Decompose
We often want to think of a function with a long rule as being the
sum, product, or quotient of other functions. For example
f (x ) = 8x 2 − 4x − 3 can be written as the sum of two functions
f1 (x ) and f2 (x ). If f1 (x ) = 8x 2 and f2 (x ) = −4x − 3, then
f (x ) = f1 (x ) + f2 (x ).
Find functions as instructed.
1 f (x ) = x 2 − x − 6 Write f (x ) as the sum of two functions
f1 (x ) and f2 (x ).
2 f (x ) = (x − 7)(x + 5) Write f (x ) as the product of two
functions f1 (x ) and f2 (x ).
3
f (x ) =
f2 (x ).
x 2 +2x +5
x −2
as the quotient of two functions f1 (x ) and
Challenge
f (x ) = x 2 − x − 6 Write f (x ) as the quotient of two functions
f1 (x ) and f2 (x ).
Division
Before dividing polynomials review how we divide numbers.
12
1
1 5
1 2
3
3
3 1 1
7 3 4
7
6
1 3
1 2
1 4
1 2
2
15734
2
= 1311 + .
12
12
Division of Polynomials
Polynomials are divided using the same algorithm.
x −2
3x 3 +4x 2 +11x
+23
3x 4 −2x 3 +3x 2
+x
−5
4
3
3x −6x
4x 3 +3x 2
4x 3 −8x 2
11x 2
+x
11x 2 −22x
23x
−5
23x −46
41
Division of Polynomials
Divide the following:
4x 3 − 3x 2 + 5x − 7
2x − 5