1. Polynomial Functions Advanced Math Chapter 3

2. Quadratic Functions and Models Advanced Math Section 3.1

3. Advanced Math 3.1 - 3.33 Quadratic function Polynomial function of degree 2

4. Advanced Math 3.1 - 3.34 Parabola “u”-shaped graph of a quadratic function May open up or down

5. Advanced Math 3.1 - 3.35 Axis of symmetry Vertical line through the center of a parabola Vertex: where the axis intersects the parabola

6. Advanced Math 3.1 - 3.36 Standard Form Convenient for sketching a parabola because it identifies the vertex as (h, k). If a > 0, the parabola opens up If a < 0, the parabola opens down

7. Advanced Math 3.1 - 3.37 Graphing a parabola in standard form Write the quadratic function in standard form by completing the square. Use standard form to find the vertex and whether it opens up or down

8. Advanced Math 3.1 - 3.38 Example Sketch the graph of the quadratic function

9. Advanced Math 3.1 - 3.39 Writing the equation of a parabola Substitute for h and k in standard form Use a given point for x and f(x) to find a

10. Advanced Math 3.1 - 3.310 Example Write the standard form of the equation of the parabola that has a vertex at (2,3) and goes through the point (0,2)

11. Advanced Math 3.1 - 3.311 Finding a maximum or a minimum Locate the vertex If a > 0, vertex is a minimum (opens up) If a < 0, vertex is a maximum (opens down)

12. Advanced Math 3.1 - 3.312 Example The profit P (in dollars) for a company that produces antivirus and system utilities software is given below, where x is the number of units sold. What sales level will yield a maximum profit?

13. Polynomial Functions of Higher Degree Advanced Math Section 3.2

14. Advanced Math 3.1 - 3.314 Polynomial functions Are continuous –No breaks –Not piecewise Have only smooth rounded curves –No sharp points This section will help you make reasonably accurate sketches of polynomial functions by hand.

15. Advanced Math 3.1 - 3.315 Power functions n is an integer greater than 0 If n is even, the graph is similar to f(x)=x 2 If n is odd, the graph is similar to f(x)=x 3 The greater n is, the skinner the graph is and the flatter the it is near the origin.

16. Advanced Math 3.1 - 3.316 Compare

17. Advanced Math 3.1 - 3.317 Compare

18. Advanced Math 3.1 - 3.318 Examples Sketch the graphs of:

19. Advanced Math 3.1 - 3.319 The Leading Coefficient Test If n (the degree) is odd The left and right go opposite directions A positive leading coefficient means the graph falls to the left and rises to the right –As x becomes more positive, the graph goes up A negative leading coefficient means the graph rises to the left and falls to the right –As x becomes more negative, the graph goes up

20. Advanced Math 3.1 - 3.320 The Leading Coefficient Test If n (the degree) is even The left and right go the same direction A positive leading coefficient means the graph rises to the left and rises to the right –The graph opens up A negative leading coefficient means the graph falls to the left and falls to the right –The graph opens down

21. Advanced Math 3.1 - 3.321 The Leading Coefficient test Does not tell you how many ups and downs there are in between See the Exploration on page 277

22. Advanced Math 3.1 - 3.322 Zeros of polynomial functions For a polynomial function of degree n There are at most n real zeros There are at most n – 1 turning points (where the graph switches between increasing and decreasing). There may be fewer of either

23. Advanced Math 3.1 - 3.323 Finding zeros Factor whenever possible Check graphically

24. Advanced Math 3.1 - 3.324 Repeated zeros

25. Advanced Math 3.1 - 3.325 Standard form For a polynomial greater than degree 2 –Terms are in descending order of exponents from left to right

26. Advanced Math 3.1 - 3.326 Graphing polynomial functions 1.Write in standard form 2.Apply leading coefficient test 3.Find the zeros 4.Plot a few additional points 5.Connect the points with smooth curves.

27. Advanced Math 3.1 - 3.327 Examples

28. Advanced Math 3.1 - 3.328 The Intermediate Value Theorem See page 282 Helps locate real zeros Find one x value at which the function is positive and another x value at which the function is negative Since the function is continuous, there must be a real zero between these two values Use the table on a calculator to get closer to the zero and approximate it

29. Advanced Math 3.1 - 3.329 Examples Use the intermediate value theorem and the table feature to approximate the real zeros of the functions. Use the zero or root feature to verify.

30. Polynomial and Synthetic Division Advanced Math Section 3.3

31. Advanced Math 3.1 - 3.331 Long division of polynomials Write the dividend in standard form Divide –Divide each term by the leading term of the divisor

32. Advanced Math 3.1 - 3.332 Examples

33. Advanced Math 3.1 - 3.333 Checking your answer Graph both the original division problem and your answer The graphs should match exactly

34. Advanced Math 3.1 - 3.334 Remainders Write remainder as a fraction with the divisor on the bottom Examples:

35. Advanced Math 3.1 - 3.335 Division Algorithm Get rid of the fraction in the remainder by multipling both sides by the denominator.

36. Advanced Math 3.1 - 3.336 Synthetic division The shortcut Works with divisors of the form x – k, where k is a constant Remember that x + k = x – (– k)

37. Advanced Math 3.1 - 3.337 Synthetic division Use an L-shaped division sign with k on the outside and the coefficients of the dividend on the inside Leave space below the dividend Add the vertical columns, then multiply diagonally by k

38. Advanced Math 3.1 - 3.338 Examples

39. Advanced Math 3.1 - 3.339 Remainder Theorem If a polynomial f(x) is divided by x – k, then the remainder is r = f(k) The remainder is the value of the function evaluated at k

40. Advanced Math 3.1 - 3.340 Examples Write the function in the form f(x) = (x – k)q(x) + r for the given value of k, and demonstrate that f(k) = r

41. Advanced Math 3.1 - 3.341 If the remainder is zero… (x – k) is a factor of the dividend (k, 0) is an x-intercept of the graph

42. Advanced Math 3.1 - 3.342 Example Show that (x + 3) and (x – 2) are factors of f(x) = 3x 3 + 2x 2 – 19x + 6. Write the complete factorization of the function List all real zeros of the function