5.1 - Polynomials. Ex: Let k(x) = x 2 +2x+1. Find (and completely simplify) the following: (a) k(1) (b) k( 2) (c) k(a)

Transcription

1 c Kathryn Bollinger, March 15, Polynomials Def: A function is a rule (process) that assigns to each element in the domain (the set of independent variables, x) ONE AND ONLY ONE element in the range (the set of dependent variables). D R Function Notation If a correspondence is a function, we use the notation f(x), read f of x or f at x. Ordered pairs of the form (x,y) can then be written as (x,f(x)). Ex: Let k(x) = x 2 +2x+1. Find (and completely simplify) the following: (a) k(1) (b) k( 2) (c) k(a) (d) k(a+b) (e) k(x+h) (f) k(x+h) k(x)

2 c Kathryn Bollinger, March 15, Interval Notation A set of real numbersthat makes upaportionof thenumberlinecan berepresentedby aninterval. A closed interval, [a,b] = a x b, is the set of all numbers between a and b, including both endpoints. An open interval, (a,b) = a < x < b, is the set of all numbers between a and b, NOT including either endpoint. If an interval includes one endpoint, but not the other, the interval is called half-open or half-closed: (a,b] or [a,b) [ ]: ( ): The union of two intervals, A B, is the set of all numbers in either interval. Ex: Represent the following using interval notation. (a) 0 x < 9 (b) All real numbers, x, except x = 1 and x = 4 (c) x > 5 and x 2 (d) x 7 and x 3,0 (e) x < 1 or x 5

3 c Kathryn Bollinger, March 15, Vertical Line Test: If any vertical line passes through two or more points on the graph of an equation, then the equation does not define a function. Ex: Determine whether each relation is a function of x. If it is, find the domain and range of the function, as well as the function values f( 2),f(0), and f(3). y x y x y x

4 c Kathryn Bollinger, March 15, Polynomial Functions Def: A polynomial function of degree n has the form f(x) = a n x n +a n 1 x n 1 + +a 1 x+a 0 where a 0,a 1,...,a n are real numbers with a n 0 and n is a non-negative integer. The coefficient a n is known as the leading coefficient of the polynomial. n = 0 : f(x) = a 0 n = 1 : f(x) = a 1 x+a 0 n = 2 : f(x) = a 2 x 2 +a 1 x+a 0 n = 3 : f(x) = a 3 x 3 +a 2 x 2 +a 1 x+a 0 n = 4 : f(x) = a 4 x 4 +a 3 x 3 +a 2 x 2 +a 1 x+a 0 Ex: Which of the following are polynomials? For each polynomial, give its degree and circle the leading coefficient. (a) f(x) = 2 x 2 5x 8 (b) g(x) = x 1/4 +x+4 (c) h(x) = 3x+x 4 π (d) k(x) = 2x 2 x (e) n(x) = 5x 3 x 2 +x 1 Odd-Degree Polynomials Even-Degree Polynomials

5 c Kathryn Bollinger, March 15, Polynomial Parent Functions You should be familiar with these basic quadratic and cubic functions, as all other quadratic and cubic functions are just shifts, stretches and reflections of these. Quadratic (Second Degree Polynomial): y = x 2 Cubic (Third Degree Polynomial): y = x 3 Determining Zeros of a Function Def: The real zeros (or roots) of a function are where f(x) = 0...its x-intercepts. Ex: Find the location of all zeros for the following polynomials. (a) f(x) = 2x+1 (b) f(x) = x 3 5x 2

6 c Kathryn Bollinger, March 15, Quadratic Functions Def: A function of the form f(x) = ax 2 +bx+c is called a quadratic function, where a,b, and c are real numbers and a 0. This is called the standard form of a quadratic function. a > 0 a < 0 Def: The maximum or minimum of a quadratic function f(x), whose graph is known as a parabola, is called its vertex and has coordinates ( ( )) b b (x,y) = (h,k) = 2a,f 2a Ex: Given f(x) = 2x 2 +4x+3, determine (a) the direction the parabola opens. (b) the vertex (is it a max or min?). (c) the maximum value and the minimum value of the function. (d) the domain of f(x). (e) the range of f(x).

7 c Kathryn Bollinger, March 15, Ex: Given f(x) = 3x 2 18x+35, determine (a) the direction the parabola opens. (b) the vertex (is it a max or min?). (c) the maximum value and the minimum value of the function. (d) the domain of f(x). (e) the range of f(x). Quadratic functions can also be written in vertex form: y = a(x h) 2 + k where a 0. The value of a is still used to determine if the parabola opens up or down, but the vertex is very easy to identify. In this form, the vertex is the point (h,k). Ex: Find the vertex and state the maximum/minimum value for the following quadratic functions. (a) y = 4(x+5) 2 3 (b) y = 3(x 4) 2 +9

8 c Kathryn Bollinger, March 15, Determining Zeros of a Quadratic Function To find the EXACT zeros of a quadratic function: 1. Factor, set each factor equal to zero, and solve for x. 2. Use the quadratic formula: x = b± b 2 4ac 2a provided b 2 4ac 0 Ex: Find the EXACT location of all zeros (x-intercepts) of the following quadratic functions: (a) f(x) = x 2 4 (b) g(x) = 6x 2 7x 3 (c) f(x) = 2x 2 +4x+3 (d) f(x) = x 2 +1 Ex: Solve x 2 +3x = 40

9 c Kathryn Bollinger, March 15, In Chapter 1 we discussed how to find a linear revenue function of a company, if the item being sold had a fixed selling price, s,= R(x) = sx. However, we also learned that the selling price of an item could be determined by consumers in the form of a price-demand function (p = mx+b). So, in general, if you are given a linear price-demand function, p, then the revenue function will be R(x) = px = (mx+b)(x) = mx 2 +bx, which will be a quadratic function. Ex: Suppose the price-demand function for a product is given by p = 7x+84. (Assume the price is given in dollars.) (a) What is the revenue function? (b) How many items shouldbesold in order tomaximize revenue? What is themaximum revenue? (c) What is the price per unit when the revenue is maximized?

10 c Kathryn Bollinger, March 15, Ex: A company sells gadgets. They can sell 10 gadgets when the price is $170 and they can sell 20 gadgets when the price is $120. The company incurs production costs of $70 per gadget and the company has fixed costs of $805. (a) Find the price-demand equation, assuming it is linear. (b) How many items should be sold in order to maximize profit? What is the maximum profit? (c) Determine the price of a gadget when the profit is maximized. (d) Determine the price of a gadget when the revenue is maximized. (e) How many gadgets does the company need to sell in order to break-even?