polynomial functions and graphs. aat-a ib - hr date: 2/25/2014 id check objective: swbat evaluate...
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Polynomial Functions and Graphs
AAT-A IB - HR Date: 2/25/2014 ID CheckObjective: SWBAT evaluate polynomial functions.Bell Ringer: Check HomeworkHW Requests: Pg 350 #16-36 evens
In class: pg 350 #17-3 by 4s 40, 42, 44, 49-52HW: pg 350 #37-52 Read Section 7.2
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Higher Degree Polynomial Functions and Graphs
an is called the leading coefficientn is the degree of the polynomiala0 is called the constant term
Polynomial FunctionA polynomial function of degree n in the variable x is a function defined by
where each ai is real, an 0, and n is a whole number.
x-Intercepts (Real Zeros)
Number Of x-Intercepts of a Polynomial FunctionA polynomial function of degree n will have a maximum of n x- intercepts (real zeros).Find all zeros of f (x) = -x4 + 4x3 - 4x2. -x4 + 4x3 - 4x2 = 0 We now have a polynomial equation. x4 - 4x3 + 4x2 = 0 Multiply both sides by -1. (optional step) x2(x2 - 4x + 4) = 0 Factor out x2. x2(x - 2)2 = 0 Factor completely.x2 = 0 or (x - 2)2 = 0 Set each factor equal to zero. x = 0 x = 2 Solve for x. (0,0) (2,0)
Polynomial FunctionsThe largest exponent within the polynomial determines the degree of the polynomial.
Polynomial Function in General FormDegreeName of Function1Linear2Quadratic3Cubic4Quartic
Polynomial Functionsf(x) = 3ConstantFunctionDegree = 0Maximum Number of Zeros: 0
f(x) = x + 2LinearFunctionDegree = 1Maximum Number of Zeros: 1Polynomial Functions
f(x) = x2 + 3x + 2QuadraticFunctionDegree = 2Maximum Number of Zeros: 2Polynomial Functions
f(x) = x3 + 4x2 + 2Cubic FunctionDegree = 3Maximum Number of Zeros: 3Polynomial Functions
Quartic FunctionDegree = 4Maximum Number of Zeros: 4Polynomial Functions
Leading CoefficientThe leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees.For example, the quartic function f(x) = -2x4 + x3 5x2 10 has a leading coefficient of -2.
Slide 2- *Leading Coefficient Test for Polynomial End BehaviorLeading Coefficient Test
leading coefficient degree of polynomial Left Right + even + ++ odd -+ - even - -- odd +-
ExampleUse the Leading Coefficient Test to determine the end behavior of the graph of f (x) = x3 + 3x2 - x - 3.
Determining End BehaviorMatch each function with its graph.
Quartic PolynomialsLook at the two graphs and discuss the questions given below.1. How can you check to see if both graphs are functions?3. What is the end behavior for each graph?4. Which graph do you think has a positive leading coeffient? Why?5. Which graph do you think has a negative leading coefficient? Why?2. How many x-intercepts do graphs A & B have?
ExtremaTurning points where the graph of a function changes from increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n 1 is at most n 1.
Local maximum point highest point or peak in an intervalfunction values at these points are called local maxima
Local minimum point lowest point or valley in an intervalfunction values at these points are called local minima
Extrema plural of extremum, includes all local maxima and local minima
Number of Local ExtremaA linear function has degree 1 and no local extrema.A quadratic function has degree 2 with one extreme point.A cubic function has degree 3 with at most two local extrema.A quartic function has degree 4 with at most three local extrema.How does this relate to the number of turning points?
The Leading Coefficient Test
The Leading Coefficient Test
Comprehensive GraphsThe most important features of the graph of a polynomial function are: intercepts,extrema,end behavior.A comprehensive graph of a polynomial function will exhibit the following features:all x-intercepts (if any),the y-intercept,all extreme points (if any),enough of the graph to exhibit end behavior.
Multiplicity and x-InterceptsIf r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.
ExampleFind the x-intercepts and multiplicity of f(x) =2(x+2)2(x-3)Zeros are at(-2,0)(3,0)
**Teachers: This definition for degree has been simplified intentionally to help students understand the concept quickly and easily.***