Lecture 6: Sections 2.2 and 2.3 Polynomial Functions, Quadratic Models

L6 - 1 Lecture 6: Sections 2.2 and 2.3 Polynomial Functions, Quadratic Models

Polynomial Functions

Def. A polynomial function of degree n is a function of the form n n−1 f(x) = anx + an−1x + ... + a1x + a0, (an =6 0) where a0, a1, ..., an are constants called coefficients and n is a nonnegative integer. an is called the leading coefficient. ex. Determine if the given function is a polynomial; if so, find its degree. 2 √ 1) f(x) = 4x8 − x6 − 4x2 + 3 5

2 2) f(x) = x3 − + 4x2 x

NOTE: The domain of a polynomial function

A polynomial function of degree 1: L6 - 2 Quadratic Functions A polynomial function of degree 2, 2 f(x) = a2x +a1x+a0, a2 =6 0, is called a quadratic function. We write f(x) = ex. Graph the following: 1) y = x2

2) y = −(x + 3)2 3) y = (x − 2)2 + 1

6 6

- - L6 - 3 Graphing a quadratic function I. Standard Form of a Quadratic Function:

We can graph a quadratic equation in standard form using translations. ex. Graph f(x) = −x2 + 6x − 6.

- L6 - 4 II. Graph using the vertex formula: We can prove the following using the Quadratic For- mula (see text) and very easily using calculus: The vertex of the graph of f(x) = ax2+bx+c is given by the formula x = and y = .

The parabola opens upwards if a > 0 and downwards if a < 0. We use the vertex and intercepts to graph. ex. Sketch the graph of y = 2x2 + 5x − 3.

- L6 - 5 Applications ex. A Florida citrus grower estimates that in his grove if 60 orange trees are planted the average yield per tree is 400 oranges. The average yield per tree will decrease by 4 oranges for each additional tree planted. Express the yield y in terms of the number of additional trees, x. How many trees should he plant to maximize the yield of his grove? L6 - 6 ex. The unit price, p, of a new model television is a function of the average number of units sold weekly, x. The ﬁnancial department of the manufacturer has observed that this relationship appears to be linear. If an average of 1000 televisions sell weekly when the price of the unit is $500, and the demand increases by 200 units per week when the price is lowered by $10, ﬁnd the demand function that models this relationship (assume 0 ≤ x ≤ 11, 000).

1) Find the revenue function, R(x). L6 - 7

2) How many televisions should be sold to maximize revenue?

3) Fixed costs of production for the television are $6500 per month, and variable costs are given by 0.01x2 + 70x dollars. Find the cost function C(x). Find the profit function P (x) which gives the total profit from the monthly sales of the television. How many televisions should be sold per month, and at what price, to maximize profit? L6 - 8 Higher Degree Polynomials

Def. A power function is a function of the form f(x) = xr for any real number r. A power function f(x) = xn, where n is a nonnegative integer, will be a polynomial.

n odd n even L6 - 9 ex. Sketch the graph of f(x) = −(x + 1)3 − 2

- L6 - 10 ex. Consider the graph of the polynomial f(x) = x4 − x3 − 4x2 + 4x = x(x − 1)(x2 − 4)

We observe the following about the graph of a polynomial of degree n (text, page 70):

• A polynomial function of degree n can have at most n − 1 turning points, and a function with n turning points must have degree n+1 or higher (to be shown later using calculus).

• The number of x-intercepts is determined by the zeroes of the function f(x) (each value of x which is a solution to the equation f(x) = 0).

• The end behavior of a polynomial function as x approaches −∞ and +∞ is determined by the highest degree term axn: the corresponding power function xn, and the sign of leading coeﬃcient a. L6 - 11 Additional Example: Supply and Demand ex. The supply and demand functions for a given product are p = S(q) = q2 + 12 and p = D(q) = 172 − 6q respectively, where p is the wholesale price per item and q is the quantity sold/produced. Find the equilibrium quantity and price.

- L6 - 12

Now You Try It!

1. Sketch the graph of the quadratic function f(x) = −x2 − 4x + 3 by writing in standard form and using translations.

2. Sketch the graph of f(x) = 3x2 − 2x − 5. Find the vertex and all intercepts.

3. A new action figure is sold at a local chain of toy stores. The district manager observes that an average of 170 figures will sell per month when the selling price is $7, but a price increase of $3 results in an average drop in sales of 30 per month. (a) Using a linear model, express price p as a function of the number of figures sold in a month, x. (b) Find the revenue function R(x). Find the price that should be charged to maximize revenue. What is the maximum revenue? (c) Fixed costs are $750 and it costs $4 to manufacture each figure. Find each production level at which the manufacturer will break even. (d) For what production levels will the manufacturer earn a profit? How many should be sold to maximize profit?

4. Farmers in a certain community are earning $8 per bushel for their potatoes on September 10, at the peak of the harvest season. But as more are brought to market, price drops by 5 cents per bushel per day. On September 10, a farmer has 140 bushels of potatoes in one field, and he estimates that the crop is increasing by an average of one bushel per day. Express the farmer’s revenue as a function of the number of days after September 10 until he harvests his crop. When should he harvest the potatoes in order to maximize his revenue? 5. Consider the polynomial sketched below. If the leading coefficient is axn, find the sign of a and the smallest possible value of n.