Homework “Mini-Quiz”10 min. (NO TALKING!!) Do NOT write the question – Answer Only!! 1)A function expressed in the form f(x) = kx a where k and a are nonzero constants is called a ________. 2)In the function above, k is called ______________. 3)Functions in the form f(x) = kx a which represent direct variations have ______ values of a whereas inverse variations have _____ values. 4)Identify the constant of proportion & power in the functions f(x) = -2/x ½. 5)For non-linear regression equations, r 2 is called the ____________ and ___ < r 2 < ___. 6)Is S = 4  r 2 a monomial? If so, state the degree and leading coefficient. If not, explain why not. Name the indep variable. 7)State the power and constant of variation for the function, graph it, and analyze f(x) = -3x 3. If you finish before the timer sounds, please remain quietNO TALKING!!

Think Pair Share Activity 1)Describe the graph of f(x) = -2x -½. 2)Is the function even, odd, or undefined for x < 0? 3)How can you identify each power function shown below as even or odd without graphing? a.f(x) = -x -3 b.g(x) = 5x 8 c.h(x) = - ½x -3/2

2.3 Polynomial Functions of Higher Degree with Modeling Graph polynomial functions Predict their end behavior Find their real zeros algebraically or graphically

The Vocabulary of Polynomials Each monomial in this sum f(x) = a n x n + a n-1 x n-1 + …+ a 2 x 2 + a 1 x + a 0 – a n x n, a n-1 x n-1,…,a 0 – is a term of the polynomial. A polynomial functions written in this way, with terms in descending degree, is written in standard form. The constants a n, a n-1,…, a 0 are the coefficients of the polynomial The term a n x n is the leading term, and a 0 is the constant term.

Ex 1 Describe how to transform the graph of an appropriate monomial function f(x) = a n x n. a)g(x) = -(x+5) 3 b)g(x) =  (x-3) 3 +1

Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most n-1 local extrema and at most n zeros. Graph each function below. Identify the number of zeros and local extrema. f(x) = x 2 f(x) = x 3 f(x) = x 3 – x 2 f(x) = x 4 + x 3 – x 2

Ex 2 Graph the polynomial function, locate its extrema and zeros, and explain how it is related to the monomials from which it is built. a)f(x) = -x 4 + 2x b)f(x) = x 3 + x 2

Do Now The end behavior of higher power functions is often related to the basic functions we have discussed Complete the exploration on p Describe the patterns you observe. In particular, how do the values of the coefficient a n and the degree n affect the end behavior of f(x)?

Leading Term Test for Polynomial End Behavior For any polynomial function f(x) = a n x n +..+a 1 x+a 0, the limits andare determined by the degree n of the polynomial and its leading coefficient a n. n Odd n even a n > 0 a n < 0

Ex 3 Graph the polynomial in a window showing its extrema and zeros and its end behavior. Describe the end behavior using limits. a)f(x) = - x 3 + 4x x – 70 b)f(x) = 2x 4 – 5x 3 – 17x x + 41

Finding Zeros is cool – just don’t make ‘em!! Ex 4 a) Find the zeros of f(x) = 3x 3 – x 2 – 2x algebraically. Ex 5 Use a graphing calculator to find the zeros of f(x) = x 5 – 10x 4 + 2x x 2 – 3x – 55.

Modeling with Power Functions Ex 6 Squares of width x are removed from a 10-cm by 25-cm piece of cardboard, and the resulting edges are folded up to form a box with no top. Determine all values of x so that the volume of the resulting box is at most 175 cm 3.

Ex 7 A state highway patrol safety division collected the data on stopping distance in the table shown. a)Draw a scatter plot of the data. b)Find the quadratic regression model. c)Sketch the graph of the function with the data points. d)Use the regression equation to predict the stopping distance for a vehicle traveling at 25 mph. e)Use the regression model to predict the speed of a car if the stopping distance is 300 ft. Highway Safety Division Speed (mph)Stopping Distance (ft)

Tonight’s Assignment P. 202 – 205 Ex 3-60 m. of 3, 66,68, 73, 74

Exit Ticket Take out notes from 9/25 and turn in before leaving!! Remember to study and have a great day!!