Investigating Graphs Of Polynomial Functions
InvestigatingGraphsof ofInvestigatingGraphsPolynomial FunctionsPolynomial FunctionsWarm UpLesson PresentationLesson QuizHoltHoltMcDougalAlgebra 2AlgebraAlgebra22HoltMcDougal
Investigating Graphs ofPolynomial FunctionsWarm UpIdentify all the real roots of each equation.1. x3 – 7x2 8x 16 03–1, 42. 2x – 14x – 12 0–1, –2, 33. x4 x3 – 25x2 – 27x 004. x4 – 26x2 25 01, –1, 5, –5Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsObjectivesUse properties of end behavior toanalyze, describe, and graphpolynomial functions.Identify and use maxima and minima ofpolynomial functions to solve problems.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsVocabularyend behaviorturning pointlocal maximumlocal minimumHolt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsPolynomial functions are classified by theirdegree. The graphs of polynomial functions areclassified by the degree of the polynomial.Each graph, based on the degree, has adistinctive shape and characteristics.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsEnd behavior is a description of the values ofthe function as x approaches infinity (x )or negative infinity (x– ). The degree andleading coefficient of a polynomial functiondetermine its end behavior. It is helpful whenyou are graphing a polynomial function toknow about the end behavior of the function.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsHolt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsExample 1: Determining End Behavior of PolynomialFunctionsIdentify the leading coefficient, degree, andend behavior.A. Q(x) –x4 6x3 – x 9The leading coefficient is –1, which is negative.The degree is 4, which is even.As x – , P(x)– , and as xB. P(x) 2x5 6x4 – x 4 , P(x)– .The leading coefficient is 2, which is positive.The degree is 5, which is odd.As x– , P(x)Holt McDougal Algebra 2– , and as x , P(x) .
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 1Identify the leading coefficient, degree, andend behavior.a. P(x) 2x5 3x2 – 4x – 1The leading coefficient is 2, which is positive.The degree is 5, which is odd.As x – , P(x)– , and as x , P(x) .b. S(x) –3x2 x 1The leading coefficient is –3, which is negative.The degree is 2, which is even.As x – , P(x)– , and as xHolt McDougal Algebra 2 , P(x)– .
Investigating Graphs ofPolynomial FunctionsExample 2A: Using Graphs to Analyze PolynomialFunctionsIdentify whether the function graphed hasan odd or even degree and a positive ornegative leading coefficient.As x– , P(x) , and as x , P(x)– .P(x) is of odd degree with a negative leading coefficient.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsExample 2B: Using Graphs to Analyze PolynomialFunctionsIdentify whether the function graphed hasan odd or even degree and a positive ornegative leading coefficient.As x– , P(x) , and as x , P(x) .P(x) is of even degree with a positive leading coefficient.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 2aIdentify whether the function graphed hasan odd or even degree and a positive ornegative leading coefficient.As x– , P(x) , and as x , P(x)– .P(x) is of odd degree with a negative leading coefficient.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 2bIdentify whether the function graphed hasan odd or even degree and a positive ornegative leading coefficient.As x– , P(x) , and as x , P(x) .P(x) is of even degree with a positive leading coefficient.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsNow that you have studied factoring, solvingpolynomial equations, and end behavior, you cangraph a polynomial function.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsExample 3: Graphing Polynomial FunctionsGraph the function. f(x) x3 4x2 x – 6.Step 1 Identify the possible rational roots by usingthe Rational Root Theorem. 1, 2, 3, 6p –6, and q 1.Step 2 Test all possible rational zeros until a zero isidentified.Test x –1.Test x 1.–114 1 –6–1 –3 213 –2 –4114115–661560x 1 is a zero, and f(x) (x – 1)(x2 5x 6).Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsExample 3 ContinuedStep 3 Write the equation in factored form.Factor: f(x) (x – 1)(x 2)(x 3)The zeros are 1, –2, and –3.Step 4 Plot other points as guidelines.f(0) –6, so the y-intercept is –6. Plot pointsbetween the zeros. Choose x – 52 , and x –1for simple calculations.f(52) 0.875, and f(–1) –4.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsExample 3 ContinuedStep 5 Identify end behavior.The degree is odd and the leadingcoefficient is positive so asx – , P(x) – , and asx , P(x) .Step 6 Sketch the graph off(x) x3 4x2 x – 6 byusing all of the informationabout f(x).Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 3aGraph the function. f(x) x3 – 2x2 – 5x 6.Step 1 Identify the possible rational roots by usingthe Rational Root Theorem. 1, 2, 3, 6p 6, and q 1.Step 2 Test all possible rational zeros until a zero isidentified.Test x –1.Test x 1.–11–2 –5–1 3621–3 –2811–2 –5 61 –1 –61–1 –60x 1 is a zero, and f(x) (x – 1)(x2 – x – 6).Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 3a ContinuedStep 3 Write the equation in factored form.Factor: f(x) (x – 1)(x 2)(x – 3)The zeros are 1, –2, and 3.Step 4 Plot other points as guidelines.f(0) 6, so the y-intercept is 6. Plot pointsbetween the zeros. Choose x –1, and x 2for simple calculations.f(–1) 8, and f(2) –4.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 3a ContinuedStep 5 Identify end behavior.The degree is odd and the leadingcoefficient is positive so asx – , P(x) – , and asx , P(x) .Step 6 Sketch the graph off(x) x3 – 2x2 – 5x 6 byusing all of the informationabout f(x).Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 3bGraph the function. f(x) –2x2 – x 6.Step 1 Identify the possible rational roots by usingthe Rational Root Theorem. 1, 2, 3, 6p 6, and q –2.Step 2 Test all possible rational zeros until a zero isidentified.Test x –2.–2–2 –1 64 –6–23 0x –2 is a zero, and f(x) (x 2)(–2x 3).Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 3b ContinuedStep 3 The equation is in factored form.Factor: f(x) (x 2)(–2x 3).The zeros are –2, and32.Step 4 Plot other points as guidelines.f(0) 6, so the y-intercept is 6. Plot pointsbetween the zeros. Choose x –1, and x 1for simple calculations.f(–1) 5, and f(1) 3.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 3b ContinuedStep 5 Identify end behavior.The degree is even and the leadingcoefficient is negative so asx – , P(x) – , and asx , P(x) – .Step 6 Sketch the graph off(x) –2x2 – x 6 by using allof the information about f(x).Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsA turning point is where a graph changes fromincreasing to decreasing or from decreasing toincreasing. A turning point corresponds to a localmaximum or minimum.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsA polynomial function of degree n has at most n – 1turning points and at most n x-intercepts. If thefunction has n distinct roots, then it has exactly n – 1turning points and exactly n x-intercepts. You can usea graphing calculator to graph and estimate maximumand minimum values.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsExample 4: Determine Maxima and Minima with aCalculatorGraph f(x) 2x3 – 18x 1 on a calculator,and estimate the local maxima and minima.Step 1 Graph.25The graph appears to have one–5local maxima and one local minima.Step 2 Find the maximum.Pressto access theCALC menu. Choose4:maximum.The localmaximum is approximately21.7846.Holt McDougal Algebra 25–25
Investigating Graphs ofPolynomial FunctionsExample 4 ContinuedGraph f(x) 2x3 – 18x 1 on a calculator,and estimate the local maxima and minima.Step 3 Find the minimum.Pressto access theCALC menu. Choose3:minimum.The local minimumis approximately –19.7846.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 4aGraph g(x) x3 – 2x – 3 on a calculator, andestimate the local maxima and minima.Step 1 Graph.5The graph appears to have one–5local maxima and one local minima.Step 2 Find the maximum.Pressto access theCALC menu. Choose4:maximum.The localmaximum is approximately–1.9113.Holt McDougal Algebra 25–5
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 4a ContinuedGraph g(x) x3 – 2x – 3 on a calculator, andestimate the local maxima and minima.Step 3 Find the minimum.Pressto access theCALC menu. Choose3:minimum.The local minimumis approximately –4.0887.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 4bGraph h(x) x4 4x2 – 6 on a calculator,and estimate the local maxima and minima.10Step 1 Graph.The graph appears to have onelocal maxima and one local minima. –10Step 2 There appears to be nomaximum.Step 3 Find the minimum.Pressto access theCALC menu. Choose3:minimum.The local minimumis –6.Holt McDougal Algebra 210–10
Investigating Graphs ofPolynomial FunctionsExample 5: Art ApplicationAn artist plans to construct an open box from a15 in. by 20 in. sheet of metal by cutting squaresfrom the corners and folding up the sides. Findthe maximum volume of the box and thecorresponding dimensions.Find a formula to represent the volume.V(x) x(15 – 2x)(20 – 2x)V lwhGraph V(x). Note that valuesof x greater than 7.5 or lessthan 0 do not make sense forthis problem.The graph has a local maximum ofabout 379.04 when x 2.83. Sothe largest open box will have dimensions of 2.83 in. by9.34 in. by 14.34 in. and a volume of 379.04 in3.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsCheck It Out! Example 5A welder plans to construct an open box from a16 ft. by 20 ft. sheet of metal by cutting squaresfrom the corners and folding up the sides. Findthe maximum volume of the box and thecorresponding dimensions.Find a formula to represent the volume.V(x) x(16 – 2x)(20 – 2x)V lwhGraph V(x). Note that valuesof x greater than 8 or lessthan 0 do not make sense forthis problem.The graph has a local maximum ofabout 420.11 when x 2.94. Sothe largest open box will have dimensions of 2.94 ft by10.12 ft by 14.12 ft and a volume of 420.11 ft3.Holt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsLesson Quiz: Part I1. Identify whether the function graphed has anodd or even degree and a positive or negativeleading coefficient.odd; positiveHolt McDougal Algebra 2
Investigating Graphs ofPolynomial FunctionsLesson Quiz: Part II2. Graph the function f(x) x3 – 3x2 – x 3.3.Estimate the local maxima and minima off(x) x3 – 15x – 2. 20.3607; –24.3607Holt McDougal Algebra 2
Polynomial Functions Investigating Graphs of Polynomial Functions Holt Algebra 2 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2. Holt McDougal Algebra 2 Investigating Graphs of Polynomial Functions Warm Up Identify all the real ro
Polynomial and Sinusoidal Functions Lesson #1: Polynomial Functions of Degrees 0,1, and 2 333 Investigating Polynomial Functions of Degree One The graphs of four polynomial functions of degree one are shown. a) The graphs shown have many characteristics in common. Make a list of the common characteristics in the space below.,1 "V - i Yi l-4-i-b .
I Using the TI-89 as a tool for investigating the following: (1) graphs of even- and odd-powered polynomial functions in factored form; (2) graphs of polynomial functions in factored form that contained odd and even multiplicities; (3) graphs of polynomial functions in factored form that contained imaginary zeros; (4) solving polynomial .
In Chapter 1, you analyzed functions and their graphs and determined whether inverse functions existed. In Chapter 2, you will: Model real-world data with polynomial functions. Use the Remainder and Factor Theorems. Find real and complex zeros of polynomial functions. Analyze and graph rational functions. Solve polynomial and rational inequalities.
Identify polynomial functions. Graph polynomial functions using tables and end behavior. Polynomial Functions Recall that a monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents. A polynomial is a monomial or a sum of monomials. A polynomial
Polynomial Functions Pages 209–210 Check for Understanding 1. A zero is the value of the variable for which a polynomial function in one variable equals zero. A root is a solution of a polynomial equation in one variable. When a polynomial function is the related function to the polynomial
Topic 3: Polynomial Functions Date Section Topic HW Due Date 3.1 Graphing Polynomial Functions 3.2 Adding, Subtracting, and Multiplying Polynomials 3.3 Polynomial Identities 3.4 Dividing Polynomials 3.5 Zeros of Polynomials Functions 3.6 Theorems About Roots of Polynomial Equations 3.7 Tran
THEOREM Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most local extrema and at most n zeros. n - 1 SECTION 2.3 Polynomial Functions of Higher Degree with Modeling 187 (a) a 3 0 (b) a 3 0 FIGURE 2.22 Graphs of four typical cubic functions: (a) two with positive and (b) two with negative leading .
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