Polynomial Functions Advanced Math Chapter 3

Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math Quadratic function Polynomial function of degree 2

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

Advanced Math Axis of symmetry Vertical line through the center of a parabola Vertex: where the axis intersects the parabola

Advanced Math 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

Advanced Math 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

Advanced Math Example Sketch the graph of the quadratic function

Advanced Math 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

Advanced Math 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)

Advanced Math 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)

Advanced Math 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?

Polynomial Functions of Higher Degree Advanced Math Section 3.2

Advanced Math 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.

Advanced Math 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.

Advanced Math Compare

Advanced Math Compare

Advanced Math Examples Sketch the graphs of:

Advanced Math 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

Advanced Math 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

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

Advanced Math 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

Advanced Math Finding zeros Factor whenever possible Check graphically

Advanced Math Repeated zeros

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

Advanced Math 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.

Advanced Math Examples

Advanced Math 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

Advanced Math 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.

Polynomial and Synthetic Division Advanced Math Section 3.3

Advanced Math Long division of polynomials Write the dividend in standard form Divide –Divide each term by the leading term of the divisor

Advanced Math Examples

Advanced Math Checking your answer Graph both the original division problem and your answer The graphs should match exactly

Advanced Math Remainders Write remainder as a fraction with the divisor on the bottom Examples:

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

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

Advanced Math 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

Advanced Math Examples

Advanced Math 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

Advanced Math 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

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

Advanced Math 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