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9-1Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
Chapter 9
Graphing
Introductory Mathematics & Statistics
9-2Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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)
9-3Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-4Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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)
9-5Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-6Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9.2 Plotting points (cont..)
Figure 9.1: A coordinate axes system for two variables,
x and y
9-7Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-8Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-9Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-10Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-11Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-12Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-13Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-14Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-15Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-16Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9.5 Break-even analysis (cont…)
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)
9-17Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9.5 Break-even analysis (cont…)
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
9-18Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9.5 Break-even analysis (cont…)
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
9-19Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9.5 Break-even analysis (cont…)
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
9-20Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-21Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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
9-22Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
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)