OSU MTH 111 - Act-#10 Rationals 1-Key (4 pages)

Math 111 Name Key Grp Rational Functions 1 GrpAct 10 Prerequisite Skills lowest terms domain restrictions Factoring Finding the degree of a polynomial Find the leading coefficient of a polynomial Key Terms rational function rational equation vertical asymptote horizontal asymptote hole radical equation Sections 4 6 4 7 Learning Outcomes Identify vertical and horizontal asymptotes as well as roots and holes for rational functions Solve rational and radical equations Translate radical expressions to rational exponents 1 A local manufacturing company has cost and revenue functions C x 31 2x 15000 and R x 69 9x where x is the number of units produced It costs 31 20 to produce one unit and the company has fixed costs of 15000 The product sells as 69 90 per unit From these we have a profit function of P x R x C x 69 9x 31 2x 15000 38 7x 15000 Total Profit Units Produced P x 38 7x 15000 For this company the average profit is given by P x x x The company calculates their average profit P as a Determine each of the following and explain what they represent in the context of this problem P 1000 23 700 P 1000 23 70 P 10000 37 20 P 100000 38 55 b The graph of P x is given Label the axes A v e c P x has a horizontal asymptote of y 38 7 Explain what this asymptote represents in the context of this problem d This function has an x intercept Give the coordinates of that x intercept and explain why it occurs in the context of the problem P r o f i t 358 0 Number of items produced This is the Break Even Point i e make fewer items and you loose money or make more and you make a profit At a production rate of 358 units the cost is equal to the revenue 2 A big part of learning about rational functions is understanding how to identify asymptotes and intercepts The last function we saw had both a horizontal and a vertical asymptote In this problem we will look a little more closely at vertical asymptotes You should have an understanding of what a vertical asymptote is and why one occurs Consider the function 3x 2 3x 6 3 x 2 x 1 f x 2 2 x 4 x 16 2 x 4 x 2 a For what values of x will the numerator of f x be zero x 2 1 b For what values of x will the denominator of f x be zero x 4 2 c What is the domain of f 2 2 4 4 d Identify any vertical asymptotes of f x 4 and x 2 e Identify the x and y intercepts of the graph of f x intercepts 2 0 and 1 0 y intercept 0 3 8 3 Now we will turn our attention towards horizontal asymptotes Horizontal asymptotes can describe the end behavior of rational functions Not all rational functions have a horizontal asymptote Decide if each of the following has a horizontal asymptote or if it will blow up down to infinity If the function has a horizontal asymptote give its equation a f x x 1 3x 5 x 2 2 H asymptote y 0 Explain how to determine if a rational function has each of the following when looking at its formula A horizontal asymptote of y 0 b g t 3t 2 5t 2 t 1 H asymptote None A horizontal asymptote other than y 0 c h x 5 x3 3x 2 1 2 x3 4 No horizontal asymptote H asymptote y 5 2 Check out the box on page 293 if you need some help Fill in the blanks with either the word numerator or denominator If a rational function f x has a vertical asymptote of x a then the denominator of f will be zero when x a If a rational function f x has a root of x a then the numerator of f will be zero when x a Note that if both the numerator and denominator are zero for the same value of x that may or may not be a vertical asymptote of the function There could merely be a hole in the graph of the function We will explore this more a little bit later Reminder Vertical asymptotes are vertical lines Reminder Horizontal asymptotes are horizontal lines They are identified by an equation of the form x a They are identified by an equation of the form y b 4 Consider the rational function g x 3x 2 6x 9 3 x 3 x 1 2x x 2 x 2 2x 3 8x a Identify the vertical asymptotes of g Draw them in on the graph as dotted lines and label V Asymptotes x 0 x 2 and x 2 b Identify the horizontal asymptotes of g Draw them in on the graph and label H asymptote y 0 c Identify all intercepts of g Label them on the graph x intercepts 3 0 1 0 y intercept None d Sketch a graph of g without a calculator