chapter 4€¦ · chapter 4 polynomial and rational functions. section 1 polynomial functions and...
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Chapter 4Polynomial and Rational
Functions
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Section 1Polynomial Functions and Models
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DAY 1
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A polynomial function is a function of the form:anxn + an-1xn-1 + … + a1x + a0
Where an – a0 are real numbersN is a non-negative integerDomain is all real numbersDegree the largest power of x that appears
Polynomial functions are the simplest algebraic expressions. They are made of repeated addition and multiplication only
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Example 1:Identify if the following are polynomials, if they are, what is their degree?
a) f(x) = 2 – 3x4 b) g(x) = √xyes, degree = 4 no (exponent needs to be an integer)
c) h(x) = x2 – 1 / x3 – 1 d) f(x) = 0no (can’t have division) yes, degree = DNE
e) g(x) = 8 f) h(x) = -2x3(x – 1)2
yes, degree = 0 yes, degree = 5
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Power Functions:
A power function of degree n is a monomial of the form
f(x) = axn
Where a is a real number, a ≠ 0, and n > o is an integer.
Examples:
f(x) = 3x f(x) = -5x2 f(x) = 8x3 f(x) = -1/2x4
D = 1 D = 2 D = 3 D = 4
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Graphs of Basic Power Functionsf(x) = xn
If n = even # If n = odd #
to y-axis Symmetry to origin
D = all real, R= y ≥ 0 Domain/Range Both all real
(-1, 1) (0, 0) (1, 1) contains points (-1, -1) (0, 0) (1, 1)
as power gets graphs as power gets bigger looksbigger, graph gets more and more verticalskinnier
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Graphing Using Transformations of Polynomials
1) Determine what the basic power function would be based on the degree in the info above to get a basic
2) Same transformation steps apply1) Horizontal shift (left/right)2) Vertical stretch/compression3) Vertical shift (up/down)
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Example 2:
Graph f(x) = 1 – x5
Odd (-1, -1) (0, 0) (1, 1)
*not adding/subtracting directly to x no right/left shift
-1 reflect (-1, 1) (0, 0) (1, -1)
+1 up 1 (-1, 2) (0, 1) (1, 0)
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Example 3:
Graph f(x) = ½(x – 1)4
Even (-1, 1) (0, 0) (1, 1)
-1 right 1 (0, 1) (1, 0) (2, 1)
½ compression (0, ½) (1, 0) (2, ½)
*nothing being added/subtracted to function no shift up/down
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EXIT SLIP
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DAY 2
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Identifying Real Zeros of a Polynomial
Zeros are the same thing as solutions, roots, and x-intercept(all found by setting the equation equal to 0 and solving for x)
Multiplicity of a zero is the exponent attached to the factor that the zero came from
Multiplicity is used to determine if the x-intercept (zero) will cross or simply touch the x-axis
If multiplicity = even # touches x-axisIf multiplicity = odd # crosses x-axis
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Example:
f(x) = x2 – 2x – 35
Zeros: 7 (multiplicity = 1 cross); -5 (multiplicity = 1 cross)
f(x) = x2 + 4x + 4
Zeros: -2 (multiplicity = 2 touch)
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To form a polynomial whose zeros, multiplicity and degree are given is following the exact reverse order of finding zeros.
Example: Find the polynomial with the given informationZeros: -1 (m = 1), 1 (m = 1), 3 (m = 1); degree: 3
Zeros: -1 (m = 1), 3 (m = 2); degree: 3
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Turning Points (local maxima and local minima)
If f is a polynomial function of degree n, then f has at most n – 1 turning points
If the graph of a polynomial function f has n – 1 turning points, the degree of f is at least n
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Example:
If f(x) = x3 + 2x2 + 1, the graph could at most have _______ turning points.
Given the following graph, there are ________ turning points. Therefore, f(x) has to at least be of degree ___________.
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End Behavior
For very large values of x, either positive or negative, the graph of the polynomial
f(x) = anxn + an-1xn-1 + … + a1x + a0
Resembles (and has an end behavior of) the graph of the power function
y = anxn
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Example:
What is the end behavior of the polynomial f(x) = -2x2(x – 2)
What is the end behavior of the polynomial f(x) = (2x + 1)(x – 3)2
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Analyzing the Graph of a Polynomial Function (pg. 185 – blue box)
Step 1: Determine the end behavior of the graph of the functionStep 2: Find the x- and y-intercepts of the graph of the functionStep 3: Determine the zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the x-axis at each interceptStep 4: Use a graphing utility to graph the functionStep 5: Approximate the turning points of the graphStep 6: Use the information in steps 1-5 to draw a complete graph of the function by handStep 7: Find the domain and range of the functionStep 8: Use the graph to determine where the function is increasing and decreasing
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Example: Use the steps to analyze f(x) = x4 – 2x3 – 8x2
Step 1:
Step 2:
Step 3: 0 multiplicity = 2 touches4 multiplicity = 1 cross-2 multiplicity = 1 cross
Step 4:
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Step 5:Min: (-1.38, -6.35) (2.88, -45.33)Max: (0, 0)Step 6: Step 7:
Step 8:
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EXIT SLIP