Download Interactive Study Guide for Students: Trigonometric Functions

Interactive Study Guide for Students
Chapter 7: Polynomial Functions
Section 1: Polynomial Functions
Polynomial Functions
Examples
A example of a ____________ in ______ variable would be r2 – 3r + State the degree of the leading
1 because it contains only one variable, r. A polynomial of degree
coefficient of each polynomial in
one variable. If it is not a
n in one variable x is an expression of the form:
polynomial in one variable,
aoxn + a1xn-1 + … + an-2x2 + an-1x + an where ao, a1,…an represent
explain why: 1. 7x4 + 5x2 + x -9
real numbers, ao is not zero, and n represent a nonnegative
integer.
2. 8x2 + 3xy -2y2
example: 3x5 + 2x4 – 5x3 + x2 + 1 where n = 5, ao = 3, a1 = 2, a2 = 5, a3 = 1, a4 = 0, a5 = 1
The __________ of a ___________ in one variable is the greatest
exponent of the variable. The _____________ ______________ is
the coefficient of the term with the highest degree. A polynomial
equation used to represent a function is called a ______________
_________________.
3. 7x6 -4x3 +
1
x
4. Find p(a2) if p(x)=x3+4x2-5x
5. Find q(a+1) -2q(a) if
q(x)=x2+3x+4
A polynomial function:
P(x) = aoxn + a1xn-1 + … + an-2x2 + an-1x + an where ao, a1,…an
represent real numbers, ao is not zero, and n represent a
nonnegative integer.
For each graph:

Describe the end
behavior.

Determine whether it
represents an odddegree or an evendegree polynomial.

State the number of real
zeros (roots).
Graphs of Polynomial Functions
Draw the general shapes of the polynomial functions below.
These graphs show the ___________ number of times the graph of
each type of polynomial may intersect the x-axis. Tell the degree
of each function.
Constant = ___
Linear =__
Quadratic = __
Cubic = __
Quartic = __
Quintic = __
6.
The _____ ___________is the behavior of the graph as x
approaches positive infinity (______) and negative infinity
(_____). This is represented as __________________, and
_______________ respectively.
Degree: even
Degree: odd
Leading coefficient: positive
Leading coefficient: positive
End behavior:
End behavior:
___________ ____________
___________ ____________
___________ ____________
___________ ____________
7.
8.
Degree: even
Degree: odd
Leading coefficient: negative
Leading coefficient: negative
End behavior:
End behavior:
___________ ____________
___________ ____________
___________ ____________
___________ ____________
Chapter 7: Polynomial Functions
Section 2: Graphing Polynomial Functions
Graph Polynomial Functions
To graph a ____________ ___________, use a _____________
_________________.
Examples
1. Graph f(x)=x4+ x3-4x2-4x
Notice the end behaviors, the roots or zeros, and use the trace button
to find them.
The ____________ __________ is used to find the roots.
If f(a)<0 and f(b)>0, then there is one root in between a and b.
How many real roots?
2. f(x)=x3-5x2+3x+2
Maximum and Minimum Points
A graph has a ________________ _______________ point if no other
nearby points have a greater y-coordinate.
Use trace to identify the
roots:
Likewise, a point is a ____________ _______________ point if no
other nearby points have a lesser y-coordinate.
3. f(x)=x3-3x2+5
These are often referred to as ___________ ___________.
The graph of a polynomial function of degree n has at most n-1
turning points.
Estimate the x-coord. of
relative max. & min. occur:
4. The average fuel consumed
between 1960 and 2000 is
f(t)= .025t3-1.5t2+18.25t+654
Chapter 7: Polynomial Functions
Section 3:Solv. Eq. using Quadratic Techniques
Quadratic Form
Examples
An expression that is in _____________ _________ can be written as
au2 + bu + c for any numbers a, b, and c, a≠0, where u is some
expression in x.
4
2
2 2
Write in quadratic form, if
possible:
1. x4 + 13x2 + 35
2
Example: x – 16x + 60 can be written (x ) – 16(x ) + 60 by letting u =
x2.
2. 16x6 – 625
3. 12x8 – x2 + 10
4. x – 9x 1 / 2 + 8
Solve.
Solve Equations Using Quadratic Form
5. x4 -13x2 +36 = 0
You can solve higher-degree polynomial equations that can be written
using _____________ ________ or have an expression that contains
a ____________ ___________.
6. x3 + 343 = 0
7. x2/3 – 6x1/3 + 5 = 0
8. x - 6 x = 7
Chapter 7: Polynomial Functions
Section 4: The Remainder and Factor Theorems
Synthetic Substitution
Examples
Synthetic ____________ (learned in Chapter 5) is a shorthand method
of long division. It can also be used to find the value of a function.
f(x) = 2x4 – 5x2 + 8x -7 1.find
f(6) using synthetic
substitution
Method 1: Long Division
Method 2: Synthetic Division
4a + 5__________
a-2
2_
4
4a2 – 3a + 6
4a2 – 8a
4
-3
6
8
10
5
16
2. find f(6) using direct
substitution
5a + 6
5a – 10
16
3. Divide
f(x)=x4+x32
17x -20x+32 by (x-4).
Compare the remainder of 16 to f(2):
f(2) = 4(2)2 – 3(2) + 6 = 16
This illustrates the _______________ ___________________ which
is: If a polynomial f(x) is divided by x-a, the remainder is the constant
f(a).
When synthetic division is used to evaluate a function, it is called
_______________ _________________. It is a convenient way of
finding the value of a function, especially when the degree of the
polynomial is greater than 2.
Factors of Polynomials
From example #4, notice that when you divide a polynomial by one of
its binomial factors, the quotient is called a __________ __________.
Since the remainder is 0, this means that (x-4) is a factor of the
polynomial. This illustrates the ____________ ____________ which is
a special case of the Remainder Theorem.
Factor Theorem: the binomial x-a is a factor of the polynomial f(x) if
and only if f(a)=0.
4. Show that x+3 is a factor of
x3+6x2-x-30. Then find the
remaining factors of the
polynomial.
5. The volume of a
rectangular prism is
v(x)=x3+3x2-36x+32. One
factor is x-4, find the missing
measures.
Chapter 7: Polynomial Functions
Section 5: Roots and Zeros
Types of Roots
Examples
Remember that a zero of a function f(x) is any value c such that f(c)
=0. When the function is graphed, the real zeros of the function are
the x-intercepts of the graph.
Solve each equation. State the
number and types of roots.
1. x + 3 = 0
n
Let f(x)= anx + … + a1x + a0 be a polynomial function, then

c is a zero of the polynomial function f(x),

x – c is a factor of the polynomial function f(x),and

c is a root or solution of the polynomial equation f(x) = 0.
2. x2 – 8x + 16 = 0
3. x3 +2x = 0
In addition, if c is a real number, then (c, 0) is an intercept of the
graph of f(x).
When you solve a polynomial equation with degree greater than zero,
it may have ______ or _____ real roots, or __ real roots (the roots are
imaginary numbers). Since real numbers and imaginary number both
belong to the set of complex numbers, then we have the
_____________ _____________ of ______________ which is that
every polynomial equation with degree greater than zero has at
______ ___ root in the set of complex numbers.
Descartes’ Rule:
The number of _______ ______ _______ of y=P(x) is number of
changes of sign of coefficients or less by an even number.
4. x4 – 1 = 0
Find the number of positive
and negative zeros:
5. p(x)=x5-6x4-3x3+7x2-8x+1
The number of _______ _______ ______ of y=P(x)is number of
changes of sign of coeff. Of P(-x) or less by even number.
Find all the zeros:
Find Zeros
6. g(x)=x3+6x2+21x+26
What strategies can we use to find the zeros?
The _____________ ______________ Theorem states that if an
imaginary is a zero, then its conjugate is also a zero. In other words, if
a+bi is a zero, the a-bi is also.
Chapter 7: Polyniomial Functions
Identify Rational Theorems
7. Write a polynomial
function of least degree with
integral coefficients whose
zeros include 3 and 2-i.
Section 6: Rational Zero Theorem
Examples
Sometimes it is not possible to test all the possible zeros of a function
using synthetic substitution. Use the _____________ __________
____________ which is:
List all the possible rational
zeros of each function:
1. f(x) = 2x3-11x2+12x+9
If f(x) = aoxn + a1xn-1 + … + an-2x2 + an-1x + an is a polynomial function
with integral coefficients, an
p
is a rational number in simplest form
q
2. f(x) = x3 - 9x2 – x + 105
and is a zero of y = f(x), then p is a factor of an and q is a factor of a0.
Example:
3
Let f(x) = 2x3 + 3x2 – 17x + 12. If
is a zero of f(x), then 3 is a factor
2
of 12 and 2 is a factor of 2. All possible zeros would be all the
combinations of the factors of 12 divided by the factors of 2, or __,
__, __, __, __, __, and __.
3. The volume of a
rectangular solid is 675 cm3.
The width is 4cm less than the
height, and the length is 6cm
more than the height. Find
the dimensions.
p
1
2
-24
Find Rational Zeros
Once you have found all the ____________ rational zeros, _________
each number using synthetic substitution.
67
5
1
3
5
9
4. Find all the zeros of
f(x)=2x4-13x3+23x2-52x+60.
p
q
2
Chapter 7: Polynomial Functions
Arithmetic Operations
-13
23
-52
60
Section 7: Operations on Functions
Examples
Operation
Definition
Examples
f(x)=x2-3x+1 g(x)=4x+5
1. Find (f+g)x
2. Find (f-g)x
f(x)=x2+5x-1 g(x)-3x-2
3. Find (fg)x
4. Find (
Composition of Functions
Suppose f and g are functions such that the _________ of g is a subset
of the _________ of f. Then the composition function f o g can be
described by the equation:
(f o g)(x) = ________________
Example using mappings: f(x)={(3,4),(2,3),(-5,0)}
5),(4,3),(0,2)}
fog
g(x)={(3,-
f
)x
g
f(x)={(7,8),(5,3),(9,8)}
g(x)={(5,7),(3,5),(7,9)}
Find f o g and g o f.
5.
gof
f(x)=x+3 g(x)=x2+x-1
6. Find (f o g)(x) and
f)(x)
Chapter 7: Polynomial Functions
Find Inverses
(g o
Section 8: Inverse Functions and Relations
Examples
Remember that a relation is a set of ordered pairs. The
___________ _____________ is the set of ordered pairs
obtained by ___________ the coordinates of each of the
original ordered pairs. The domain becomes the
_________ and the range becomes the ___________.
Example: Q = {(1, 2), (3, 4), (5, 6)}
{(2, 1), (4, 3), (6, 5)}
___________ relations.
S=
Q and S are
1. The vertices of a right triangle are
(2,
1), (5, 1) and (2, -4). Graph the triangle, find
the vertices of the inverse relation, and
determine if the resulting triangle is also a
right triangle. Can you find the equation of
the line that the triangle reflects across to
get the inverse?
The ordered pairs of __________ _____________ are
also related and can be defined as f(x) and f-1(x). Suppose
f and f-1 are inverse functions, then f(a) = b iff f-1(b) = a. In
simple terms, change the x and the y and solve for y.
2. Find the inverse of f(x) =
x6
2
Inverses of Relations and Functions
You can determine whether two functions are inverses by
finding both of their _______________. If both are the
___________ function, then they are inverses.
3. Determine whether f(x)=5x+10 and
1
5
g(x)= x–2 are inverse functions.
[f o g](x) = x and [g o f] (x) = x
When the inverse of a ___________ is a _________, then
the original function is said to be
_____-__-_____.
To determine if the inverse of a function is a function, you
can use the horizontal line test.
Chapter 7: Polynomial Functions
Square Root Functions
Section 9: Square Root Functions and Inequalities
Examples
If a function contains a square root of a variable, it is called a
__________ _______ ____________. The inverse of a
quadratic function is a square root function only if the
_________ is restricted to _____-_________ numbers.
Calculator activity: (use the 2nd
1. Graph y = 3x  4 . What is the
domain? What is the range?
x-intercept?
y-intercept?
buttons)
1. Set window to [-2,8] &[-4,6]. Graph y= x , y= x + 1 and
y= x - 2. State domain and range for each function,
describe similarities and differences.
2. Set window to [0,10][0,10]. Graph y= x , y= 2 x and
y= 8 x . State domain and range for each function, describe
similarities and differences.
3. Set window to [-10,10][-10,10].How would you write an
equation that translates the parent graph to the left three
units. Test your equation. What did you get?
Square Root Inequalities
2. A lookout on a submarine is h feet
above the surface of the water. The
greatest distance d in miles that the
lookout can see on a clear day is given
by the square root of the quantity h
multiplied by
3
. Graph the function,
2
state the domain and range.
A square root inequality is and ___________ (<,<,>,>)
involving __________ _________. Use what you know
about graphing inequalities.
4. Graph y<
2x  6
5. Graph y> x  1
3. A ship is 3 miles from the submarine.
How high should the periscope be to see
it?