Example 4. The daily cost of manufacturing a particular product is given by where x is the number of units produced each day. Determine how many units should be produced daily to minimize cost. Algebraic Solution Graphical Solution Producing 35 units per day will minimize cost. Mo’ Money, Long Problems – Warm Up C(x) = 1200 – 7x + 0.1x 2

Announcements Assignment p. 228 #10, 12, 13, 14, 20 Test Corrections After School Today and Tomorrow Please come in after school for tutoring for Friday’s Quiz

Assignment Questions? p. 214 # 32, 34, 37, 38, 39, 44, 70

Section Objectives: Students will know how to sketch and analyze graphs of polynomial functions. 2.2 Polynomial Functions of a Higher Degree

Graphs of Polynomial Functions

Characteristics of the graphs of polynomial functions. 1. Polynomial functions are continuous. This means that the graphs of polynomial functions have no breaks, holes, or gaps. 2. The graphs of polynomial functions have only nice smooth turns and bends. There are no sharp turns.

Fight the Power We will first look at the simplest polynomials, f(x) = x n. We can break these into two cases, n is even and n is odd. Odd. What do we notice? Even. What do we notice?

Identify the Highest Degree We have to identify the greatest degree/power/exponent of the equation

The Leading Coefficient Test – Left and Right Hand Behavior f(x) = a n x n +  Positive coefficient a n > 0 Negative Coefficient a n < 0 Highest degree is even Highest degree is odd Depending on the highest degree in the polynomial, the graph will behave in different ways.

f(x) = a n x n +  a n > 0a n < 0 n even n odd

Example 2. Describe the right-hand and left-hand behavior of the graph of each function. a) f(x) = -x 4 + 7x 3 – 14x - 9 Falls to the left and right b) g(x) = 5x 5 + 2x 3 – 14x Falls to the left and rises to the right

Warm - Up Factoring End behavior g(x) = -5x 5 + 2x 3 – 14x Rises to the left and falls to the right

Announcements Assignment: p. 228 # 27, 30, 32, 36, 39, 48, 52 Test corrections and quiz review after school Objective Find out how to find and use zeros of polynomial functions as sketching aids

Assignment Questions p. 228 #10, 12, 13, 14, 20

Zeros of Polynomial Functions f is a polynomial function and a is a real number. 1. x = a is a zero of f. 1. Where y is equal to 0 2. x = a is a solution of the equation f(x) = Setting f(x) = 0 will give us our solutions 3. (x – a) is a factor of f(x). 1. Factoring f(x) will give us the solutions 4. (a, 0) is an x-intercept of the graph of f. 1. Solving for x will give us our x-intercepts

Factoring??

Example 3. Find the x-intercept of the graph of f(x) = x 3 – x 2 – x + 1. Algebraic Solution The x-intercepts are (-1, 0) and (1, 0). Graphical Solution Note that 1 is a repeated zero. In general, a factor (x – a) k, k > 1, yields a repeated zero x = a of multiplicity k. If k is odd, the graph crosses the x-axis at x = a. If k is even, the graph only touches the x-axis at x = a.

It’s all in the graphs 1. Locate the “x-intercepts” or zeros of the polynomial function. 2. Determine the multiplicity of each zero 1. If the function crosses through, the multiplicity is odd 2. If the function touches, the multiplicity is even 3. Write down the “factored form” of the polynomial

Example 4. State the equation of the following graph. 1. Since the x-intercepts are (0, 0) and (2, 0). Also, 0 has multiplicity 2, therefore the graph will just touch at (0, 0) 2. The graph will rise to right and fall to the left. f(x) = x 2 (x – 2).

The Intermediate Value Theorem Used to estimate zeros If there are two points on a line, and the line crosses in between those two points, then the x – intercept will be an x – value between the x – values of the two points

Summary How are we able to find zeros of polynomial functions? How are we able to use zeros of polynomial functions as sketching aids?