SECTION 5.1: Polynomials

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1 1 SECTION 5.1: Polynomials Functions Definitions: Function, Independent Variable, Dependent Variable, Domain, and Range A function is a rule that assigns to each input value x exactly output value y = f(x). The variable x is called the variable and y is called the variable. The is the set of allowable x values and the is the set of all possible y values. Example 1: Graph of a Function The graph of a function y = f(x) is shown below. a) Find f( 2) and f(0). b) Determine the values of x that make f x = 4. Vertical Line Test A curve in the xy-plane is the graph of a if and only if any vertical line passes through the curve at most. In other words, if any vertical line passes through a curve or more times, then the curve is a function.

2 2 Example 2: Applying the Vertical Line Test Determine whether each curve is a function of x. a) b) c)

3 3 Interval Notation We will be using to be describing the domain and range of our functions. means all numbers between a and b, including a and b. means all numbers between a and b, including a and b. means all numbers between a and b, not including a but including b. means all numbers between a and b, including a but not including b. Example 3: Finding Domain and Range of a Function Find the domain and range of the following function. Example 4: Function Notation Let f x = x! 3x + 1. Calculate the following: a) f( 1) b) f( ) c) f(x + h)

4 4 Quadratic Polynomials Definition: Quadratic Polynomial A function is a function of the form f x = ax! + bx + c, where a, b, and c are real numbers and a 0. The form f x = ax! + bx + c is called standard form. The graph of a quadratic function f(x) is a called a. The highest or lowest point of a parabola is called the of the quadratic function. It is the point h, k, where h =!! and k = f(h) (the quadratic function evaluated at h)!! Example 5: Finding the Vertex, Max/Min, Domain and Range Given the quadratic function y = f x = 3x! + 6x + 1, a) Find the vertex b) Find the maximum c) Find the minimum d) Find the domain and range.

5 5 Definition: Zeros The (or roots) of a quadratic function are its x-intercepts. Given the quadratic function y = f x = ax! + bx + c, you have 3 different options for calculating its zeros. 1. Factor. 2. Quadratic Formula: x = b ± b! 4ac 2a 3. Calculator: Example 6: Calculating the Vertex and the Zeros of a Quadratic Letting y = f x = 3x! 4x + 3, a) Find the vertex b) Is the vertex a maximum or a minimum of f? c) Find the zeros of f.

6 6 Remark: A quadratic function f x = ax! + bx + c can have 0, 1, or 2 zeros. If, then you have NO zeros. If, then you have ONE zero. If, then you have TWO zeros. Applications of Quadratic Polynomials: Quadratic Revenue & Profit Before we described revenue using a. That is, if p was the selling price and x was the number of items sold, then revenue would be:. This assumed a constant price p. But, when we studying demand, we learned that prices could vary based on the number of items sold. Therefore, if price were given by a line p x, then revenue would be: Notice that R(x) would now be a quadratic function. Example 7: Demand and Quadratic Revenue A cellphone company determined that consumers demand 100 cellphones when the price of one phone is $440, and that consumers demand 50 cellphones when the price of one phone is $940. Assuming that the demand function is linear and the selling price is determined by demand, a) Find the demand equation. b) Find the revenue function. c) How many phones should be sold to maximize revenue? What is the maximum revenue?

7 7 d) What is the unit price of a phone when revenue is maximized? e) If the company has fixed costs of $9000 and a variable cost of $80 per phone, find the company s linear cost function. f) Find the company s profit function. g) What is the company s maximum profit? h) How many toasters should be sold in order for the company to break-even?

8 8 Polynomials Definition: Polynomial A of degree n is a function of the form: where, the coefficients are real numbers, and the number n is a nonnegative integer. The coefficient is called the leading coefficient. We can still consider the zeros of a polynomial, although there is no general formula for finding these zeros. However, there are special cases where you can find the zeros by hand by factoring. Example 8: Zeros of Polynomials Find the zeros of the following polynomials: a) f x = x! 2x! 3x b) f x = (x! 16)(x! + 2x + 1)