Polynomial Functions Mini-Quiz

1. Homework “Mini-Quiz”10 min. (NO TALKING!!)Do NOT write the question – Answer Only!! • A function expressed in the form f(x) = kxawhere k and a are nonzero constants is called a ________. • In the function above, k is called ______________. • Functions in the form f(x) = kxa which represent direct variations have ______ values of a whereas inverse variations have _____ values. • Identify the constant of proportion & power in the functions f(x) = -2/x½. • For non-linear regression equations, r2 is called the ____________ and ___ <r2< ___. • Is S = 4pr2 a monomial? If so, state the degree and leading coefficient. If not, explain why not. Name the indep variable. • State the power and constant of variation for the function, graph it, and analyze f(x) = -3x3. If you finish before the timer sounds, please remain quietNO TALKING!!

2. Think Pair Share Activity • Describe the graph of f(x) = -2x-½ . • Is the function even, odd, or undefined for x < 0? • How can you identify each power function shown below as even or odd without graphing? • f(x) = -x-3 • g(x) = 5x8 • h(x) = - ½x-3/2

3. 2.3 Polynomial Functions of Higher Degree with Modeling Graph polynomial functions Predict their end behavior Find their real zeros algebraically or graphically

4. The Vocabulary of Polynomials • Each monomial in this sum f(x) = anxn + an-1xn-1 + …+ a2x2 + a1x + a0– anxn , an-1xn-1,…,a0 – 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 an, an-1,…, a0 are the coefficients of the polynomial • The term anxn is the leading term, and a0 is the constant term.

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

6. 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) = x2 f(x) = x3 f(x) = x3 – x2 f(x) = x4 + x3 – x2

7. 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. • f(x) = -x4 + 2x • f(x) = x3 + x2

8. Do Now • The end behavior of higher power functions is often related to the basic functions we have discussed • Complete the exploration on p. 196. • Describe the patterns you observe. • In particular, how do the values of the coefficient an and the degree n affect the end behavior of f(x)?

9. Leading Term Test for Polynomial End Behavior For any polynomial function f(x) = anxn+..+a1x+a0, the limits and are determined by the degree n of the polynomial and its leading coefficient an. n even an < 0 an > 0 an > 0 an < 0 n Odd

10. Ex 3 Graph the polynomial in a window showing its extrema and zeros and its end behavior. Describe the end behavior using limits. • f(x) = - x3 + 4x2 + 31x – 70 • f(x) = 2x4 – 5x3 – 17x2 + 14x + 41

11. Finding Zeros is cool – just don’t make ‘em!!  Ex 4 a) Find the zeros of f(x) = 3x3 – x2 – 2x algebraically. Ex 5 Use a graphing calculator to find the zeros of f(x) = x5 – 10x4 + 2x3 + 64x2 – 3x – 55.

12. 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 cm3.

13. Ex 7 A state highway patrol safety division collected the data on stopping distance in the table shown. • Draw a scatter plot of the data. • Find the quadratic regression model. • Sketch the graph of the function with the data points. • Use the regression equation to predict the stopping distance for a vehicle traveling at 25 mph. • Use the regression model to predict the speed of a car if the stopping distance is 300 ft.

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

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