Hello,

I need help working through my math 055 course activity, that is due 4/12/2017. Help would be appreciated!

Percent Markup

Objectives

1. Determine the domain of

a rational function defined

by an equation of the form

k

y =

, where k is

g1x2

a nonzero constant and

g1x2 is a first-degree

polynomial.

2. Identify the vertical and

horizontal asymptotes of

k

y =

.

g1x2

3. Sketch graphs of rational

functions defined by

k

y =

.

g1x2 Percent Markup 501 You are a buyer for a national chain of retail stores. You purchase merchandise at a wholesale

cost. The merchandise is then sold at a retail price (called the selling price). The retailers

markup is the difference between the selling price (what the consumer pays) and the wholesale cost.

1. a. You acquire a line of sports jackets at a wholesale cost of $80 per jacket. If the jackets sell for $120 each at the retail level, what is the amount of the markup? b. The markup is what percent of the selling price? (This percent is called the percent

markup of the selling price.) 2. The relationship among the selling price, S, the wholesale cost, C, and the percent

markup, P, of the selling price (expressed as a decimal) is modeled by

S = C

.

1-P If the wholesale cost of a sports jacket is $80, write an equation for S in terms of P. 3. a. Complete the following table for S =

P (% markup) 0 0.01 0.05 80

.

1-P

0.10 0.25 0.50 0.75 0.95 S (selling price) ($) b. As the values of P approach 1, what happens to the values of S? What does this

mean in practical terms? 4. a. Can the percent markup of the selling price be 100% (i.e., can P = 1)? Explain. b. What is the practical domain of this function? Copyright 2013 Pearson Education, Inc. M05_CONS8612_04_SE_C05_01.qxd 502 10/8/10 11:10 PM Page 502 Rational and Radical Functions Chapter 5 80

. Use the table of data pairs in Problem

1 - P

3 to help determine an appropriate scale. Verify your sketch using your graphing calculator. 5. Sketch a graph of the function defined by S = Graphs of f 1x2 k

g1x2 Because S and P represent real-world quantities, the practical domain limits our investigation

80

of the function defined by S =

.

1 - P

80

6. Consider the general function f defined by f 1x2 =

.

1 - x

a. What is the domain of function f? b. Complete the following table.

x - 10 -5 0 0.50 0.75 0.90 1 1.10 1.25 f(x) c. Sketch a graph of the function

f. Use the table in part b to determine an appropriate scale. Copyright 2013 Pearson Education, Inc. 1.50 2 5 10 M05_CONS8612_04_SE_C05_01.qxd 10/14/10 4:42 PM Page 503 Activity 5.3 Percent Markup 503 d. Does the graph of f have a horizontal asymptote? Explain why or why not. If you

answer yes, what is the equation of the horizontal asymptote? e. Does the graph of f have a vertical asymptote? Explain why or why not. If you

answer yes, what is the equation of the vertical asymptote? f. For what value of x is f (x) maximum? g. Does the graph have any intercepts? 7. a. Sketch a graph of f 1x2 = 80

using your graphing calculator. The screens should

1-x appear as follows. b. The vertical asymptote 1x = 12 appears to be part of the graph. What do you think

has happened? The calculator setting in Problem 7 was for connected mode. In this mode, it will connect all

the points in plots. If the window is such that the calculator does not try to plot a point at

x = 1, then it may connect a point for an x-value slightly less than 1 to one slightly greater

than 1. This creates the appearance of an asymptote. To avoid this, change the mode to dot

mode, as follows: Note: Some TI-83/TI-84 Plus calculators will not display a vertical asymptote even in connected mode. It depends on the operating system in the calculator, and there is no way to tell

simply by looking at the calculator.

8. Consider the function defined by g1x2 =

a. What is the domain of g? Copyright 2013 Pearson Education, Inc. 80

.

x + 1 M05_CONS8612_04_SE_C05_01.qxd 504 Chapter 5 10/14/10 4:42 PM Page 504 Rational and Radical Functions b. Construct a table of data points for g.

-4 x

-3 -2 -1 0 1 2 g(x) c. Sketch a graph of g using an appropriate scale. d. Determine the equation of the vertical asymptote. As a graphing aid, if the vertical

asymptote is not the y-axis 1x = 02, the asymptote is drawn as a dotted vertical line.

If you have not done so, draw the vertical asymptote in the graph in part c. e. Determine the equation of the horizontal asymptote. f. Does the graph of g have any intercepts? g. Sketch a graph of g1x2 = 80

using your graphing calculator. Your screens

1 + x should appear as follows. 9. How are the graphs of f 1x2 = 80

80

and g1x2 =

similar? How are they different?

1-x

x + 1 Copyright 2013 Pearson Education, Inc. M05_CONS8612_04_SE_C05_01.qxd 10/8/10 11:10 PM Page 505 Activity 5.3 Percent Markup 505 Rational Functions

A function Q, defined by an equation of the form

Q1x2 = k

,

g1x2 where k is a nonzero constant and g(x) is a polynomial with degree 1 and g1x2 Z 0,

belongs to the family of functions known as rational functions. The inverse variation function in Activity 5.2 is a special case of a rational function, where g1x2 = x n. The only values

at which the function is not defined are any values for which the denominator is zero.

If g1a2 = 0, then x = a is a vertical asymptote of the graph of Q = k

.

g1x2 The horizontal asymptote of the graph of function Q is the x-axis 1y = 02. Example 1

a. f 1x2 = Determine the domain and the vertical and horizontal asymptotes for

each of the following rational functions.

3

x b. g1x2 = 10

x - 4 c. h1x2 = -5

2x + 6 SOLUTION FUNCTION VERTICAL

ASYMPTOTE DOMAIN HORIZONTAL

ASYMPTOTE a. f 1x2 = 3

x all real numbers

except x = 0 x = 0 y = 0 b. g1x2 = 10

x - 4 all real numbers

except x = 4 x = 4 y = 0 c. h1x2 = -5

2x + 6 all real numbers

except x = - 3 x = -3 y = 0 10. Consider the function defined by f 1x2 = 5

.

3x - 6 a. Determine the domain of f. b. Complete the following table.

x - 10 -5 0 1.5 1.9 2 f(x) Copyright 2013 Pearson Education, Inc. 2.1 2.5 3 8 13 M05_CONS8612_04_SE_C05_01.qxd 506 Chapter 5 10/8/10 11:10 PM Page 506 Rational and Radical Functions c. Determine the vertical and horizontal asymptotes of the graph of f. d. Sketch a graph of f. e. Verify the graph using your graphing calculator. 11. Without using your graphing calculator, match the following functions with the accompanying graphs. Use your graphing calculator to verify your matches.

i. f 1x2 =

iv. F 1x2 = 5

x ii. g1x2 = 5

x2 5

x + 2 v. G1x2 = 5

2x - 4 a. b. d. e. Copyright 2013 Pearson Education, Inc. iii. h1x2 = c. -5

x M05_CONS8612_04_SE_C05_01.qxd 10/8/10 11:10 PM Page 507 Activity 5.3 Percent Markup SUMMARY: ACTIVITY 5.3

1. A function Q, defined by an equation of the form

Q1x2 = k

,

g1x2 where k is any nonzero constant, g(x) is a polynomial with degree 1 and g1x2 Z 0,

belongs to a family of functions known as rational functions.

10

10

-5

, and h1x2 =

.

Examples include f 1x2 = 2 , g1x2 =

x - 4

2x + 6

x

2. The domain of Q is the set of all real numbers except those values of the input x such that

g1x2 = 0.

k

, k Z 0.

3. If g1a2 = 0, then x = a is a vertical asymptote of Q1x2 =

g1x2

4. The horizontal asymptote is the x-axis 1y = 02. EXERCISES: ACTIVITY 5.3

1. To obtain an estimate of the required volume, V, of timber that must be harvested for a logging

company to break even, use the model

V =

where Y + L

,

P - S - F - T V is the required annual logging volume (in cubic meters),

Y is the yard cost (in dollars),

L is the loading cost (in dollars),

P is the selling price (in dollars per cubic meter),

S is the skidding cost (in dollars per cubic meter),

F is the felling cost (in dollars per cubic meter), and

T is the transportation cost (in dollars per cubic meter). a. Suppose a logging company estimates that the yard cost will be $25,000, the loading cost

will be $55,000, the skidding cost will be $1.50 per cubic meter, the felling cost will be

$0.40 per cubic meter, and the transportation cost will be $0.60 per cubic meter. Write a rule

for V as a function of P. b. Complete the following table using the equation in part a.

P ($) 2.50 3.00 5.00 10.00 25.00 V (m3) Exercise numbers appearing in color are answered in the Selected Answers appendix. Copyright 2013 Pearson Education, Inc. 507 M05_CONS8612_04_SE_C05_01.qxd 508 Chapter 5 10/9/10 2:28 AM Page 508 Rational and Radical Functions c. As the selling price per cubic meter increases, what happens to the corresponding required

logging volume, V? d. Determine the value of V when P = 2. What is the practical meaning of the

negative value of V? e. What is the practical domain of this function? f. Sketch a graph of the function. Use the table in part b to determine an appropriate scale. 2. Two functions are defined by f 1x2 = 10

10

.

and g1x2 =

x - 5

5 - x a. Describe how you can determine the vertical asymptote without graphing. b. Determine the vertical and horizontal asymptote for the graph of each function. c. Verify your answers by graphing each function on your graphing calculator. Copyright 2013 Pearson Education, Inc. M05_CONS8612_04_SE_C05_01.qxd 10/14/10 4:42 PM Page 509 Activity 5.3 Percent Markup 3. Without graphing, determine the domain of each of the following functions.

Then determine the equation of the vertical asymptote for each function. Verify your

answers using your graphing calculator.

a. f 1x2 = 6

x - 7 b. g1x2 = 20

25 - x c. h1x2 = 3

2x - 10 d. F 1x2 = 13

0.5x - 7 e. G1x2 = -4

2x + 5 4. Give examples of two different rational functions that have a vertical asymptote at x = 10. 5. As the input value of a rational function gets closer to a vertical asymptote, the output becomes

larger in magnitude, approaching either positive or negative infinity. Consider the functions

10

10

and g1x2 =

f 1x2 =

.

x - 5

5 - x

a. Determine the equations of the vertical asymptotes for functions f and g. b. Describe what happens to the output value when x is near the vertical asymptote

but to the right of it. c. Describe what happens to the output value when x is near the vertical asymptote

but to the left of it. Copyright 2013 Pearson Education, Inc. 509