Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
CAS Elementary Functions
Unit 2
Polynomial, Power and Rational Functions
Name:
Section
Due Date
2.1
Nov. 1
2.2
Nov. 6
2.3
Nov. 8
2.4
Nov. 13
2.5
Nov. 15
2.6 outline
Exploration
Additional examples with teacherled discussion of sign charts
Nov. 17
Nov. 21
Nov. 21, due 25
2.7
Nov. 27
2.8
Dec. 5
1
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
Name:
Date:
CAS Elementary Functions
Section 2.1 Part 1– Linear and Quadratic Functions
OUTLINE Pages 170 - 176
Polynomial Functions
Definition: Polynomial Function –
Give two examples of functions, one which IS a polynomial and one which IS NOT a
polynomial and explain each example.
Name
Zero
Table of Polynomial Functions of Low Degree
Form
Degree
Example
f(x)=0
Undef.
y=0
Constant
Linear
Quadratic
Linear Functions
What is the average rate of change of a function f(x)?
2
CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
What is the distinguishing characteristic of a linear function (The Constant Rate of Change
Theorem)?
What do we call the rate of change of a linear function?
Complete Exploration 1 on page 173 here. Show all work.
1.
2.
3.
4.
Definition:
Initial Value –
Constant Term –
3
CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
Linear Function Summary
Representation
Description
Verbal
Algebraic
Graphical
Analytic (Numerical)
Assignment: Section 2.1 Exercises 1 – 7
4
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
Name:
Date:
CAS Elementary Functions
Section 2.1 Part 2 – Linear and Quadratic Functions
OUTLINE Pages 176 - 181
Quadratic Functions
Definition: Quadratic Function –
What is the effect of the sign of the a coefficient in a quadratic function on the graph of the
function? Illustrate:
Definitions: Axis of Symmetry (axis, line of symmetry) –
Vertex –
Standard Quadratic Form –
Vertex Form –
5
CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
What are the relationships between a, b, and c in a quadratic equation in standard form, and a, h,
and k in a quadratic equation in vertex form?
Skip . . .
Find the vertex and axis of symmetry for the graph of f x   2x 2  12x  13 . Graph the function
and rewrite the equation in vertex form.
Quadratic Function Summary
Representation
Description
Verbal
Algebraic
Graphical
Assignment: Section 2.1, exercises 13 – 18 , 21, 40
6
CAS Elementary Functions
Name:
Definition:
Unit 2 – Polynomial, Power and Rational Functions
Date:
CAS Elementary Functions
Section 2.2 – Power Functions
OUTLINE
Power Function (Identify all constants in the definition) –
Give an example of a commonly known formula which is a power function of direct variation:
Give an example of a commonly known formula which is a power function of inverse variation:
Sketch a graph and give a complete analysis (like those given on page 189) of the following
function:
1
f x  
x
7
CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
Definition – Monomial Function:
Complete Exploration 1 from page 190 in your notebook. Draw sketches of all relevant graphs
and record your conclusions. Be prepared to show the complete exploration at the time this
outline is checked.
Explain the property of monomial functions being illustrated by figure 2.11 on page 190.
What happens in the graph when x > 1?
Summarize the symmetry of monomial functions.
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CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
Summarize and illustrate the graphical behavior of power functions (p. 192)
Use the properties and strategies provided in this section to sketch graphs of the functions below.
Explain how you were able to determine the shape of each power function. NO
CALCULATOR!
2
a. f  x   3x 3
b. f x   2 x

3
4
Assignment: 2.2 – [#3-30 multiples of 3 and 4], 33, 34, 35, 37 – 44, 46, 47, 49, 50, 54,
55 (on calc), 57 (on calc)
9
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
Name:
Date:
CAS Elementary Functions
Section 2.3 – Higher Degree Polynomial Functions
OUTLINE
Definitions: Cubic Function –
Quartic Functions –
Term (of a polynomial) –
Standard Form (of a polynomial function) –
Note in example 2 that the graph of a polynomial function near the value x = 0 acts very much
like the linear function of its final two terms ( a1x 1  a0 ). Illustrate and describe this phenomenon
1
by graphing the function f x   x 3 2x 2  2x 1 and the associated linear function
2
gx   2x 1 in each of the following windows:
[-5, 5] by [-10, 5]
[-.5, .5] by [-3, 1]
[-.1, .1] by [-1.5, .5]
Explanation/Analysis:
10
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
How many local extrema can a polynomial function have?
How many roots (zeros) can a polynomial function have?
Complete Exploration 1 parts 1 and 2 from page 203 in your notebook. Then draw the graphs of
each of the following functions and describe the end behavior of each function.
f x   2x 3  3x 2
f x   x 5  6x 2  1
1 4
3
x  x  x 2
2
1
1
f x    x 6  x 5  2x 3  2
5
4
What conclusions can you reach about the end behavior of polynomial functions?
f x  
Compare your conclusions with those summarized in the box on page 204.
What is the connection between real-number zeros, x-intercepts and solutions for a function f(x)?
Find all the roots (zeros) of
f x   x 4  x 2 algebraically:
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CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
Describe “repeated” zeros or “multiplicity” of roots of polynomial functions.
Generalize and illustrate:
Roots (Zeros) of Odd Multiplicity –
Roots (Zeros) of Even Multiplicity –
Summarize the Intermediate Value Theorem.
12
CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
Polynomial Interpolation
Complete the Exploration on page 208 here.
What conclusion is suggested by this exploration?
Assignment: 2.3 Exercises: 4, 5, 7 – 12, [18 – 28 mult. of 3 and 4], 35 – 44, 47, 50, 54, 55, 67
13
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
Name:
Date:
CAS Elementary Functions
Section 2.4 – Real Zeros of Polynomial Functions
OUTLINE
What quantities are involved in any type of division?
Summarize these quantities in an equation:
Generalize the equation you wrote above for polynomial division (green box page 214).
Rewrite the generalization in fraction form:
Demonstrate polynomial long division on the following quotient:
2x 5  x 4  x 3  3x 2  2x  4
x 3  x 2  4x  1
14
CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
The Remainder Theorem:
Use the Remainder Theorem to find the remainder when 𝑓(𝑥) = 4𝑥 3 − 8𝑥 2 + 3𝑥 − 5 is divided
by (𝑥 − 2).
When does the Remainder Theorem provide us with useful information about the roots,
intercepts and factors of a polynomial function?
Factor Theorem:
Summarize the relationship between solutions, roots, zeros, x-intercepts and factors when
discussing a polynomial function.
15
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
Demonstrate synthetic division by dividing (𝑥) = 2𝑥 4 − 3𝑥 3 + 𝑥 − 1 by 𝑥 − 2. Express the
answer in the form qx   r .
d x 
Summarize the Rational Roots (Zeros) Theorem.
Summarize the tests for upper and lower bounds for real roots of a polynomial function.
Assignment: Section 2.4 Set 1: [#3 – 25 mult. of 3, 4 and 5], 27, 28, 32
Set 2: #33 – 35, [39 – 55 mult. of 3, 4 and 5], 59 – 61
16
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
Name:
Date:
CAS Elementary Functions
Section 2.5 – Complex Zeros of Polynomial Functions
OUTLINE
Fundamental Theorem of Algebra:
Linear Factorization Theorem:
If a polynomial function with real coefficients has a root of a + bi what other complex number
must also be a root of that function?
Why does this have to be true?
Find the standard form of a 5th degree polynomial with real coefficients whose roots include
x  0, x  1 (multiplicity 2), and x  2 i .
17
CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
Summarize the principles involved in factoring a polynomial with real coefficients into factors
with real coefficients.
What must be true of a polynomial function of odd degree with real number coefficients? Why?
Assignment: Section 2.5 Exercises #3 – 42 mult. of 3 and 4
18
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
Name:
Date:
CAS Elementary Functions
Section 2.6– Graphs of Rational Functions
Part 1 OUTLINE
Definition: Rational Function –
1
Graph the function 𝑓(𝑥) = 𝑥 without using your graphing calculator. Perform a complete
analysis of this function. Include all the elements described in the list below.
a. Sketch of the graph with some indication
of scale
b. Domain:
c. Range:
d. Describe the continuity:
e. Increasing/decreasing:
f. Boundedness
g. Local Extrema
h. Asymptotes
i. End Behavior
j. Intercepts
k. Even/Odd Symmetry (where relevant, confirm this algebraically)
19
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
Describe the basic sequence of transformations that can be used to produce the graph of the
1
given function from the rational function 𝑓(𝑥) = 𝑥 , and then sketch a graph of the function.
1
−3
a. 𝑔(𝑥) = 𝑥−3 + 4
b. ℎ(𝑥) = 𝑥+1 − 2
How can any function in the form 𝑓(𝑥) =
1
𝑎𝑥+𝑏
𝑐𝑥+𝑑
, 𝑐 ≠ 0 be written in the transformed form of
a rational function 𝑓(𝑥) = 𝐴 (𝐵(𝑥+𝐶)) + 𝐷?
Extension: Prove algebraically that any linear polynomial divided by another linear polynomial
1
can be written as a transformation of the rational function in the form of 𝐴 (𝐵(𝑥+𝐶)) + 𝐷.
20
CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
Describe the basic sequence of transformations that can be used to produce the graph of the
1
given function from the rational function 𝑓(𝑥) = 𝑥 , and then sketch a graph of the function.
a. 𝑔(𝑥) =
Name:
4𝑥+1
𝑥−1
b. ℎ(𝑥) =
3𝑥−7
𝑥−2
Assignment: Section 2.6 Exercises #6-10, 31-36
Per:
Date:
21
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
CAS Elementary Functions
Rational Functions – Exploration
Without using your graphing calculator, perform a complete functional analysis on the following
function. Fully justify each property or behavior. Construct an accurate graph of the function
and completely explain why the graph correctly represents the function.
f x  
Property/Behavior
2 x 3  x 2  8x  5
x 3  x 2  5x  3
Justification
Domain:
x-intercept(s):
y-intercept:
Asymptotes:
(After completing these first four,
you can begin to construct your
graph)
Range:
Continuity:
Boundedness:
Local Extrema:
Increasing/Decreasing Behaviors:
22
CAS Elementary Functions
Unit 2 – Polynomial, Power and Rational Functions
Graph:
Explanation:
23
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
Name:
Date:
CAS Elementary Functions
Section 2.7 - Solving Equations in One Variable
OUTLINE Pages 248 - 255
Define: Rational Equations –
Extraneous Solutions –
Solve the equation:
8
1. 𝑥 − 𝑥 = 2
8
3
2. 1 − 𝑥−5 = 𝑥
24
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
3.
6
8𝑥 2
4𝑥
= 𝑥 2 −9 − 𝑥+3
𝑥−3
4. How much pure acid must be added to 50 mL of a 35% acid solution to produce a
mixture that is 75% acid?
Name:
Section 2.7: [#3-18 and 24-30 multiples of 3 and 4], 32, 34, 35, 52-55
Date:
25
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
CAS Elementary Functions
Section 2.8 - Solving Inequalities in One Variable
OUTLINE Pages 257 - 267
What method do we have for solving non-linear inequalities?
Solve the inequality:
1. (𝑥 − 3)(𝑥 + 2)2 (𝑥 + 4) ≥ 0
2. 2𝑥 3 − 5𝑥 2 + 5 > 2𝑥
26
Unit 2 – Polynomial, Power and Rational Functions
CAS Elementary Functions
3.
4.
5
𝑥+3
3
> − 𝑥−1
(𝑥−2)|𝑥−5|
√𝑥+4(𝑥+1)
≤0
Section 2.8: #3-5, 10-11, 17-19, 24, 29-31, 40-42, 49-51
27