Download 2-1 Power and Radical Functions

PreCalculus
Section 2-1 Power and Radical Functions
Name: ___________________
Period: _____
A Power Function is any function in the form f  x   ax n , where a and n are nonzero real numbers.
Power functions are monomial functions. A monomial function is any function that can be written as
f(x) = a or f  x   ax n , where a and n are nonzero constant real numbers.
Look at the key concept on page 86 for Monomial Functions.
Ex. 1: Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity
and where the function is increasing or decreasing.
1
a. f  x   x 6
b. f  x    x5
2
When n < 0, power functions, such as f  x   x1 , f  x   x2 , and f  x   x3 , are undefined at x=0.
Thus the graphs of these functions will contain discontinuities.
Ex. 2: Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity
and where the function is increasing or decreasing.
a. f  x   2 x4
b. f  x   2 x3
PreCalculus Ch 2A Notes_Page 1
p
1
p
is in simplest form, indicates the nth root of x p .
n
If n is an even integer, then the domain must be restricted to nonnegative values.
 if n is even, domain is ___________
1

n
* Domain of f  x   x
if n is odd, domain is ____________
if n is negative, domain is _________

Recall that x n indicates the nth root of x, and x n , where
Ex. 3: Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity
and where the function is increasing or decreasing.
5
a. f  x   x 4
b. f  x   4 x

2
5
Ex. 4: The following data represent the body length L in centimeters and the mass M in kilograms of
several African Golden cats being studied by a scientist.
L
M
72
11
72
12
73
13
74
15
75
15
76
14
78
15
79
15
80
14
83
16
84
16
85
15
86
17
88
17
89
18
90
18
a. Create a scatter plot of the data.
b. Determine a power function to model the data.
c. Use the data to predict the mass of an African Golden cat with a length of 77 centimeters.
PreCalculus Ch 2A Notes_Page 2
Exponential Form
x
p
n
Radical Form
n
=
xp
A radical function is a function that can be written in as f  x   n x p , where n and p are positive integers
greater than 1 that have no common factors. Look at the key concept on page 90 for Radical Functions.
* Radical function: f  x  
p
n
x
 n xp
 if n is even, domain is ___________

if n is odd, domain is ____________
Ex. 5: Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity
and where the function is increasing or decreasing.
1
a. f  x   5 2 x3
b. f  x   5 3x  4
2
To solve Radical equations:
1. Isolate the radical.
2. Raise each side of the equation to a power that is the reciprocal of the current power.
3. Solve the equation and check for extraneous solutions when the index of the radical is
even number (for example: , 4 , 6 ).
Ex. 6: Solve each equation.
a. 2 x  28x  29  3
PreCalculus Ch 2A Notes_Page 3
b. 12  3  x  2  8
2
c.
x  1  1  2 x  12
PreCalculus
Section 2-2 Polynomial Functions
*Polynomial Function:
Name: ___________________
f  x   an xn  an1xn1    a1x  a0
where an  0 and an is the leading coefficient, n is the degree, and a0 is the constant term.
The exponents of polynomial functions are all whole numbers (0, 1, 2, 3 ...)
A polynomial function is in standard form if its terms are written in descending order.
*Graphs of polynomial functions (page 97): The degree of a polynomial is the highest exponent of the
polynomial. The leading coefficient is the coefficient of that term. These can tell you about the end
behavior of the graph.
Constant function
(degree = 0)
Linear function
(degree = 1)
Ex. 1: Graph each function.
5
a. f  x    x  3
Quadratic function
(degree = 2)
b. f  x   x6  1
*End behavior of polynomial function graph: Look at the key concept on page 98.
Even-degree
(+)
leading
coefficient
(-)
leading
coefficient
PreCalculus Ch 2A Notes_Page 4
Odd-degree
Ex. 2: Describe the end behavior of the graph of each polynomial function using limits. Explain your
reasoning using the leading term test.
a. f  x   3x4  x3  x2  x  1
b. f  x   3x2  2 x5  x3
c. f  x   2x5 1
*Cubic and Quartic Functions:
Cubic function
(degree = 3)
Quartic function
(degree = 4)
Notice a cubic function has at most 3 x-intercepts (or zeros) and a quartic function has at most 4.
Turning points indicate where the graph changes direction.
Maximum and minimum points are located on turning points.
Notice a cubic function has at most 2 turning points and a quartic function has at most 3.
*A polynomial function of degree n has at most n distinct real zeros and
at most (n – 1) turning points.
* Repeated Zeros of Polynomial Functions: See the key concept on page 102.
If a factor  x  c  occurs more than once in the completely factored form of f(x), then it is called a
repeated zero. If  x  c  is a factor of polynomial function f, then c is a zero of multiplicity m of f,
where m is a natural number.
m
Ex. 3: State the number of possible real zeros and turning points of f  x   x3  5x2  4x . Then
determine all of the real zeros by factoring.
Ex. 4: State the number of possible real zeros and turning points of h  x   x4  4x2  3 . Then determine
all of the real zeros by factoring.
PreCalculus Ch 2A Notes_Page 5
Ex. 5: State the number of possible real zeros and turning points of h  x   x4  5x3  6 x2 . Then
determine all of the real zeros by factoring and state the multiplicity of any repeated zeros.
To graph a polynomial function:
1. Apply the leading-term test to determine end behavior.
2. Determine the zeros and state the multiplicity of any repeated zeros
3. Find a few additional points (choose x-values that fall in the intervals determined by zeros)
4. Connect the dots and sketch the graph
Ex. 6: For f  x   x  3x  1 x  2  , (a) apply the leading-term test, (b) determine the zeros and state the
multiplicity of any repeated zeros, (c) find a few additional points, and then (d) graph the function.
2
You Try: Graph the polynomial function, completing all steps.
3
a) h  x   2 x  x  4  3x  1
b) g  x    x3  2 x2  8x
PreCalculus Ch 2A Notes_Page 6
Ex. 7: The table below shows a town’s population over an 8-year period. Year 1 refers to the year 2001,
year 2 refers to 2002, and so on.
1
Year
Population 5050
2
3
5510 5608
4
5496
5
5201
6
5089
7
5095
a. Create a scatter plot of the data, and determine the type of polynomial function that could be used
to represent the data.
b. Write a polynomial function to model the data set. Round each coefficient to the nearest
hundredth.
c. Use the model to estimate the population of the town in the year 2010.
d. Use the model to determine the approximate year in which the population reaches 10,169.
PreCalculus Ch 2A Notes_Page 7
PreCalculus
Section 2-3 The Remainder and Factor Theorems
Name: ___________________
*When you divide a polynomial f(x) by a divisor d(x), you get a quotient q(x) and a remainder r.
f  x
d  x
 q  x 
r
d  x
. If r  0 , then
f  x
d  x
 q  x  or f ( x)  d ( x)  q( x)
Consider the polynomial f  x   6x3  25x2  18x  9 . If you know that f has a zero at x = 3, then you
know that  x  3 is a factor of f. Because f is a third-degree polynomial, you know that there exists a
second-degree polynomial factor q  x  such that f  x    x  3  q  x  . Solving this equation for q, it’s
true that q  x  
f  x
 x  3
. To divide polynomials, use long division or synthetic division.
Ex. 1: Factor 6 x3  17 x 2  104 x  60 completely using long division if  2 x  5 is a factor.
You Try: Factor each polynomial completely using division and the given factor.
a) x3  7 x 2  4 x  12 ; x  6
b) 6 x3  2 x 2  16 x  8 ; 2 x  4
* How to set up polynomial long division: See key concept on page 111.
1) Write the dividend and divisor in descending powers of the variable.
2) Insert placeholders with zero coefficients for missing terms.
Synthetic Division can be used to divide any polynomial by a divisor in the form x – c.
PreCalculus Ch 2A Notes_Page 8
Ex. 2: Divide 6 x3  5 x 2  9 x  6 by 2x – 1.
Ex. 4: Divide using synthetic division.
a.  2 x5  4 x 4  3x3  5 x  8    x  3
Ex. 3: Divide x3  x 2  14 x  4 by x 2  6 .
b.  8 x 4  38 x3  5 x 2  3x  3   4 x  1
You Try: Divide using long division and then synthetic division.
 6 x4  11x3  15x2  12 x  7   3x  1
*Remainder Theorem:
If a polynomial f(x) is divided by (x – c), then the remainder is r  f (c) .
*Factor Theorem:
A polynomial f(x) has a factor (x – c) if and only if f (c)  0 .
PreCalculus Ch 2A Notes_Page 9
Synthetic Substitution: The Remainder Theorem states that to evaluate a function f  x  for x  c ,
you can divide f  x  by x  c using synthetic division. The remainder will be f  c  .
Using synthetic division to evaluate a function is called synthetic substitution.
Ex. 5: Evaluate the given function for the given value, using synthetic division.
f  x   x6  2x5  4x4  2x3  8x  3; c  2
Ex. 6: Suppose 800 units of beachfront property have tenants paying $600 per week. Research indicates
that for each $10 decrease in rent, 15 more units would be rented. The weekly revenue from the rentals is
given by R  x   150x2  1000x  480,000 , where x is the number of $10 decreases the property
manager is willing to take. Use the Remainder Theorem to find the revenue from the properties if the
property manager decreases the rent by $50.
Ex. 7: Use the Factor Theorem to determine if the binomials given are factors of f(x). Use the binomials
that are factors to write a factored form of f(x).
a. f  x   x3 18x2  60 x  25;  x  5 ,  x  5
b. f  x   x3  2x2 13x  10;  x  5 ,  x  2
PreCalculus Ch 2A Notes_Page 10