Introductory Mathematics
& Statistics
Chapter 9
Graphing
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9-1
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)
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9-2
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
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9-3
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)
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9-4
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
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PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9-5
9.2 Plotting points (cont..)
Figure 9.1: A coordinate axes system for two variables,
x and y
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9-6
9.3 Plotting a straight line
• A linear equation is one that may be written in one of the
following forms:
y  bx  a
or
y  a  bx
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
Copyright  2010 McGraw-Hill Australia Pty Ltd
PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9-7
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
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9-8
9.3 Plotting a straight line (cont…)
Example
Plot on a graph the line of the equation y = 2x + 3
Solution
x -value
y -value
0
3
2
7
-2
-1
-4
-5
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PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9-9
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
Copyright  2010 McGraw-Hill Australia Pty Ltd
PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9-10
9.4 Solving simultaneous equations with
the aid of a graph (cont…)
Example
Plot the following equations
Solution
x -value
y -value
x -value
y -value
3x + 4y = 33
-3
0
5
10.5 8.25
4.5
2x -3y = 5
-5
0
-5 -1.67
4
1
8
2.25
10
5
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9-11
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
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9-12
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
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9-13
9.5 Break-even analysis (cont…)
Total cost formula
Total cost  variable cost  fixed cost
or
C  vx  f
Where
x = number of items manufactured
v = variable cost to manufacture each item
f = fixed cost of manufacture
C = total cost
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9-14
9.5 Break-even analysis (cont…)
• Income
– Total income formula
I  sx
– 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
P IC
– If the value of P is negative, it represents a loss
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9-15
9.5 Break-even analysis (cont…)
Example
A 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)
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9.5 Break-even analysis (cont…)
Solution
v  $8,
(a)
f  $500, s  $12
C  vx  j
 $8  80  $500  $1140
Hence, the total cost of manufacturing 80 calculators in a
week is $1140.
(b)
I  sx
 $12  80  $960
Hence, the income from selling 80 calculators is $960.
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9-17
9.5 Break-even analysis (cont…)
Solution (cont…)
(c)
P  IC
 $960  $1140  $180
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
C  8x  500
and
I  12x
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9-18
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 per
week, with the total cost and income each equaling $1500
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9-19
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
y  x2
y  6  x2
y  2x 2  4 x  6
y x
•
To plot non-linear graphs, we can simply plot as many points
as necessary until we obtain the general shape of the curve
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9-20
9.6 Non-linear graphs and turning points
(cont…)
Example
Draw the graph that represents the equation
Solution
x -value
y -value
0
8
0.5
8.75
1
9
1.5
8.75
2
8
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2.5
6.75
y  8  2x  x 2
3
5
3.5
2.75
4
0
9-21
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)
Copyright  2010 McGraw-Hill Australia Pty Ltd
PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
9-22