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

2. 9-2 Copyright  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)

3. 9-3 Copyright  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

4. 9-4 Copyright  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)

5. 9-5 Copyright  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

6. 9-6 Copyright  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

7. 9-7 Copyright  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 or

8. 9-8 Copyright  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. 9-9 Copyright  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 Solution

10. 9-10 Copyright  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

11. 9-11 Copyright  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…) Example Plot the following equations Solution

12. 9-12 Copyright  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

13. 9-13 Copyright  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

14. 9-14 Copyright  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 Where x = number of items manufactured v = variable cost to manufacture each item f = fixed cost of manufacture C = total cost

15. 9-15 Copyright  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

16. 9-16 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 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)

17. 9-17 Copyright  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.

18. 9-18 Copyright  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

19. 9-19 Copyright  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 per week, with the total cost and income each equaling $1500

20. 9-20 Copyright  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

21. 9-21 Copyright  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…) Example Draw the graph that represents the equation Solution

22. 9-22 Copyright  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)