Chapter 7: Polynomial Functions Objectives: Evaluate polynomial functions 2)Identify general shapes of graphs of polynomial functions

Polynomial Functions Recall that a polynomial is a monomial or sum of monomials. The expression 3x2 – 3x + 1 is a polynomial in one variable since it only contains one variable, x. A polynomial of degree n in one variable x is an expression of the form a0xn + a1xn-1 + … + an-1x + an where the coefficients a0, a1, a2…an represent real numbers, a0 is not zero, and n represents a nonnegative integer. Examples: 3x5 + 2x4 - 5x3 + x2 + 1 n=5, a0=3, a1=2, a2= -5, a3=1, a4=0, and a5=1

Vocabulary Degree of a polynomial in one variable is the greatest exponent of its variable. The leading coefficient is the coefficient of the term with the highest degree. Polynomial Expression Degree Leading Coefficient Constant 9 Linear x - 2 1 Quadratic 3x2 + 4x - 5 2 3 Cubic 4x3 - 6 4 General a0xn + a1xn-1 + … + an-1x + an n a0

Example 1 State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 7x4 + 5x2 + x – 9 8x2 + 3xy – 2y2 7x6 – 4x3 + x2 + 2x3 – x5

Polynomial Function A polynomial equation used to represent a function. Examples: f(x) = 4x2 – 5x + 2 is a quadratic polynomial function. f(x) = 2x3 + 4x2 – 5x + 7 is a cubic polynomial function. If you know an element in the domain of any polynomial function, you can find the corresponding value in the range. Recall that f(3) can be found by evaluating the function for x = 3.

Example 2 a) Show that the polynomial function f(r)=3r2 – 3r + 1 gives the total number of hexagons when r = 1, 2, and 3. Find the total number of hexagons in a honeycomb with 12 rings.

Example 3 Functional Values of Variables Find p(a2) if p(x) = x3 + 4x2 – 5x Find q(a +1) – 2q(a) if q(x) = x2 + 3x + 4

Graphs of Polynomial Functions These graphs show the MAXIMUM number of times the graph of each type of polynomial may intersect the x-axis. The x-coordinate of the point at which the graph intersects the x-axis is called a zero of a function. How does the degree compare to the maximum number of real zeros? Notice the shapes of the graphs for even-degree polynomial functions and odd- degree polynomial functions. The degree and leading coefficient of a polynomial function determine the graph’s end behavior.

Partner Practice Text p. 350 #s 1-11 all

Homework Text p. 350-351 #s 16-38 even

End Behavior Is the behavior of a graph as x approaches positive infinity (+ ) or negative infinity ( ). This is represented as x + and x -

Note on x-intercepts: Graph of an even-degree function: may or may not intersect the x-axis if it intersects the x-axis in two places, the function has two real zeros. if it does not intersect the x-axis the roots of the related equation are imaginary and cannot be determined from the graph If the graph is tangent to the x-axis, as shown above, there are two zeros that are the same number

Note on x-intercepts (ct’d): Graph of an odd-degree function: ALWAYS intersect the x-axis at least once Thus, ALWAYS has at least one real zero.

Example 4a 1) describe end behavior 2) determine whether it represents an odd-degree or even-degree polynomial function 3) state the number of real zeros

Example 4b 1) describe end behavior 2) determine whether it represents an odd-degree or even-degree polynomial function 3) state the number of real zeros

Example 4c 1) describe end behavior 2) determine whether it represents an odd-degree or even-degree polynomial function 3) state the number of real zeros

Text p. 350 #s 12-15 all Text p. 351 #s 39-48 all Homework Text p. 350 #s 12-15 all Text p. 351 #s 39-48 all