9-1 copyright 2010 mcgraw-hill australia pty ltd powerpoint slides to accompany croucher,...

9-1Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

Chapter 9

Graphing

Introductory Mathematics & Statistics


Learning Objectives

• Plot ordered pairs on a graph

• Plot and interpret straight-line graphs

• Solve simple simultaneous equations using graphs

• Use simultaneous equations to solve problems in break-even analysis

• Draw and interpret non-linear graphs (including turning points)


9.1 Introduction

• One way of illustrating relationships that occur between variables is by means of a graph

• On other occasions we may be presented with information that is already in graphical form, and we need to interpret the graph

• An understanding of the basic ideas concerning graphs is invaluable to the interpretation of such displays


9.2 Plotting points

• We often have a pair of observations that are matched, e.g.– sales and year– height and weight– profit and sales– exports and imports– expenditure and income

• These quantities are called ordered pairs of observations

• The first member of the ordered pair is usually referred to as the x-coordinate and the second member as the y-coordinate

• The notation for an ordered pair of values x and y is (x, y)


9.2 Plotting points (cont..)

• Ordered pairs of observations may be plotted onto a two-dimensional plane

• In this plane we draw two perpendicular lines (called coordinate axes)– The horizontal axis it called the x-axis– The vertical axis is called the y-axis

• The point of intersection of these axes is called the origin

• On each of the axes there is a scale


9.2 Plotting points (cont..)

Figure 9.1: A coordinate axes system for two variables,

x and y


9.3 Plotting a straight line

• A linear equation is one that may be written in one of the following forms:

where a and b are constants

• The constant b is called the slope or gradient of the line, because it represents the rate at which y changes with respect to x

• The constant a represents the y-intercept, that is the value of y where the line crosses the y-axis

abxy bxay or


9.3 Plotting a straight line (cont…)

• To draw a line, plot a minimum of two points that satisfy the equation and draw the straight line that passes through them

• The points on that line will then represent all points whose coordinates satisfy the equation of the line

• It is appropriate to write the equation of the line on the line itself

• It does not matter which points on the line are plotted, as long as they satisfy the equation


9.3 Plotting a straight line (cont…)

Example

Plot on a graph the line of the equation y = 2x + 3

Solutionx -value 0 2 -2 -4y -value 3 7 -1 -5


9.4 Solving simultaneous equations with the aid of a graph

• Simultaneous equations may be solved by plotting each equation on the same diagram, then finding the coordinates of the point of intersection

• The x-coordinate and y-coordinate represent the solution to the equations

• When the two lines being plotted have the same slope, they are parallel and thus never intersect

• In this case, the simultaneous equations have no solution


9.4 Solving simultaneous equations with the aid of a graph (cont…)

ExamplePlot the following equations

Solution

x -value -3 0 5 8y -value 10.5 8.25 4.5 2.25

3x + 4y = 33

x -value -5 0 4 10y -value -5 -1.67 1 5

2x -3y = 5


9.5 Break-even analysis

• In manufacturing situations, it is good to find the number of items where the income gained exactly equals the cost of manufacturing them

• This process is known as break-even analysis and is performed either by solving a pair of simultaneous equations or with the aid of a graph

• Consider the graphical solution; this process consists of drawing one line for costs and another line for income on the same diagram and finding their point of intersection

• This point represents the break-even point


9.5 Break-even analysis (cont…)• Costs

– Costs can be classified as either fixed or variable– Fixed costs are costs that are considered independent of the

number of items produced, e.g. rent maintenance administration depreciation salaries telephone

– Variable costs are a function of the number produced, e.g. insurance labour materials


9.5 Break-even analysis (cont…)

Total cost formula

or

Wherex = number of items manufacturedv = variable cost to manufacture each itemf = fixed cost of manufactureC = total cost

costfixedcostvariablecostTotal

fvxC


9.5 Break-even analysis (cont…)• Income

– Total income formula

– Where

S = income made from each item

I = total income– There is no y-intercept term, so the line will pass through the origin

• The total Profit (P) made will be

– If the value of P is negative, it represents a loss

sxI

CIP



ExampleA company manufactures an inexpensive model of scientific calculator. There is a weekly fixed cost of $500 for producing the calculators and a variable cost of $8 per calculator. The company receives an income of $12 for each calculator that it sells.

(a) Find the total cost of manufacturing 80 calculators in a week

(b) Find the income from selling 80 calculators

(c) Find the profit (or loss) if the company manufactures and sells 80 calculators in a particular week

(d) With the aid of a graph, find the point at which total cost is equal to income (the break-even point)



Solution

(a)

Hence, the total cost of manufacturing 80 calculators in a week is $1140.

(b)

Hence, the income from selling 80 calculators is $960.

12$s,500$f,8$v

1140$500$808$

jvxC

960$8012$

sxI



Solution (cont…)

(c)

Since this value of P is negative, this represents a loss to the company of $180

(d) Suppose x = the number of calculators sold in a week, then

180$1140$960$

CIP

x12I

and

500x8C



Solution (d) (cont…)

Break-even is at the point of intersection, which is (125, 1500).Therefore, the break-even point of sales is 125 calculators perweek, with the total cost and income each equaling $1500


9.6 Non-linear graphs and turning points

• On some occasions we may be interested in graphs that are not straight lines

• Such graphs are called nonlinear and involve equations that have powers of the x-variable other than 1

• Examples of equations

• To plot non-linear graphs, we can simply plot as many points as necessary until we obtain the general shape of the curve

xy

6x4x2y

x6y

xy

2

2

2


9.6 Non-linear graphs and turning points (cont…)

ExampleDraw the graph that represents the equation

Solution x -value 0 0.5 1 1.5 2 2.5 3 3.5 4y -value 8 8.75 9 8.75 8 6.75 5 2.75 0

2xx28y


Summary

• We looked at plotting ordered pairs on a graph

• We also plotted and interpreted straight-line graphs

• We solved simple simultaneous equations using graphs

• We used simultaneous equations to solve problems in break-even analysis

• Lastly we drew and interpreted non-linear graphs (including turning points)

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