Warm-Up 5 minutes Graph each function. Describe its general shape. 1) f(x) = x3 – 4x2 – 2x + 8 2) g(x) = x4 – 10x2 + 9 3) h(x) = -x4 + 2x3 + 13x2 – 14x - 24

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

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 1 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 and x > 0.8

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 a < 0 left right n is even n is odd rise rise fall fall fall rise rise fall

Example 2 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

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 10 714 15 822 20 836 25 739 30 817 35 945 a) Find a quartic regression model for the number of students who participated in the ACT program during the given years A(x) = -0.004x4 + 0.44x3 – 17.96x2 + 293.83x - 836 b) Use the regression model to estimate the number of students who participated in the ACT program in 1985. estimate using model is about 767,000

Homework Pages