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7.2 Polynomial Functions and Their Graphs Objectives: •Identify and describe the important features of the graph of a polynomial function •Use a polynomial function to model real-world data Exploring End Behavior of f(x) = axn Graph each function separately. For each function, answer parts a-c. 1) y = x2 2) y = x4 3) y = 2x2 4) y = 2x4 5) y = x3 6) y = x5 7) y = 2x3 8) y = 2x5 9) y = -x2 10) y = -x4 11) y = -2x2 12) y = -2x4 13) y = -x3 14) y = -x5 15) y = -2x3 16) y = -2x5 a. Is the degree of the function even or odd? b. Is the leading coefficient positive or negative? c. Does the graph rise or fall on the left? on the right? Exploring End Behavior of f(x) = axn Rise or Fall??? a > 0 left a < 0 right left right n is even rise rise fall fall n is odd fall rise rise fall Example 1 Describe the end behavior of each function. a) V(x) = x3 – 2x2 – 5x + 3 falls on the left and rises on the right b) R(x) = 1 + x – x2 – x3 + 2x4 rises on the left and the right Graphs of Polynomial Functions f(a) is a local maximum if there is an interval around a such that f(a) > f(x) for all values of x in the interval, where x = a. f(a) is a local minimum if there is an interval around a such that f(a) < f(x) for all values of x in the interval, where x = a. Increasing and Decreasing Functions Let x1 and x2 be numbers in the domain of a function, f. The function f is increasing over an open interval if for every x1 < x2 in the interval, f(x1) < f(x2). The function f is decreasing over an open interval if for every x1 < x2 in the interval, f(x1) > f(x2). Example 2 Graph P(x) = -2x3 – x2 + 5x + 6. a) Approximate any local maxima or minima to the nearest tenth. minimum: (-1.1,2.0) maximum: (0.8,8.3) b) Find the intervals over which the function is increasing and decreasing. increasing: x > -1.1 and x < 0.8 decreasing: x < -1.1 or x > 0.8 Example 3 The table below gives the number of students who participated in the ACT program during selected years from 1970 to 1995. The variable x represents the number of years since 1960, and y represents the number of participants in thousands. x y a) Find a quartic regression model for the 10 714 number of students who participated in 15 822 the ACT program during the given years 20 836 A(x) = -0.004x4 + 0.44x3 – 17.96x2 + 293.83x - 836 25 739 b) Use the regression model to estimate 30 817 the number of students who participated in the ACT program in 1985. 35 estimate using model is about 767,000 945 Homework Lesson 7.2 exercises 29-36