1. STROUD Worked examples and exercises are in the text PROGRAMME F4 GRAPHS

2. STROUD Worked examples and exercises are in the text Graphs of equations Using a spreadsheet Inequalities Absolute values Programme F4: Graphs

3. STROUD Worked examples and exercises are in the text Graphs of equations Using a spreadsheet Inequalities Absolute values Programme F4: Graphs

4. STROUD Worked examples and exercises are in the text Graphs of equations Equations Ordered pairs of numbers Cartesian axes Drawing a graph Programme F4: Graphs

5. STROUD Worked examples and exercises are in the text Graphs of equations Equations Programme F4: Graphs A conditional equation is a statement of the equality of two expressions that is only true for restricted values of the symbols involved. An equation in a single variable can be written as a subject variable (called the dependent variable) being equal to some expression in the single variable (called the independent variable).

6. STROUD Worked examples and exercises are in the text Graphs of equations Ordered pairs of numbers Programme F4: Graphs Evaluating an equation of a single independent variable enables a collection of ordered pairs of numbers to be constructed. It is called an ordered pair because the first number of the pair is always the value of the independent variable and the second number is the corresponding value of the dependent variable.

7. STROUD Worked examples and exercises are in the text Graphs of equations Cartesian axes Programme F4: Graphs If, on a sheet of graph paper, two straight lines are drawn perpendicular to each other and on each line the integers are marked off so that the two lines intersect at their common zero points, then an ordered pair of numbers can be plotted as a point in the plane referenced against the integers on the two lines. This is called the Cartesian coordinate frame and each line is called an axis.

8. STROUD Worked examples and exercises are in the text Graphs of equations Drawing a graph Programme F4: Graphs If, for an equation in a single independent variable a collection of ordered pairs of points is constructed and each pair is plotted in the same Cartesian coordinate frame a collection of isolated points is obtained.

9. STROUD Worked examples and exercises are in the text Graphs of equations Drawing a graph Programme F4: Graphs It is not possible to plot every single point as there is an infinity of them. Instead, the isolated points are joined up with a continuous line known as the graph of the equation.

10. STROUD Worked examples and exercises are in the text Graphs of equations Using a spreadsheet Inequalities Absolute values Programme F4: Graphs

11. STROUD Worked examples and exercises are in the text Using a spreadsheet Spreadsheets Rows and columns Text and number entry Formulas Clearing entries Construction of a Cartesian graph Programme F4: Graphs

12. STROUD Worked examples and exercises are in the text Using a spreadsheet Spreadsheets Programme F4: Graphs Electronic spreadsheets provide extensive graphing capabilities and their use is widespread. All descriptions here are based on the Microsoft spreadsheet Excel 97 for Windows.

13. STROUD Worked examples and exercises are in the text Using a spreadsheet Rows and columns Programme F4: Graphs Every electronic spreadsheet consists of a collection of cells arranged in a regular array of columns and rows. To enable the identification of an individual cell each cell has an address given by a column label followed by a row label.

14. STROUD Worked examples and exercises are in the text Using a spreadsheet Text and number entry Programme F4: Graphs Every cell on the spreadsheet is capable of having numbers or text entered into it via the keyboard.

15. STROUD Worked examples and exercises are in the text Using a spreadsheet Formulas Programme F4: Graphs As well as text and numbers, each cell is capable of containing a formula. In an Excel spreadsheet every formula begins with the = (equals) sign when it is entered at the keyboard. For example, the formula: =3*C15 entered into a cell will ensure that the contents of the cell are 3 times the contents of cell C15 (* stands for multiplication).

16. STROUD Worked examples and exercises are in the text Using a spreadsheet Clearing entries Programme F4: Graphs To clear an entry, point and click at the cell to be cleared to make it the active cell. Click the Edit command on the Command Bar to reveal a drop- down menu. Select Clear to reveal a further drop-down menu. Select All from this menu.

17. STROUD Worked examples and exercises are in the text Using a spreadsheet Construction of a Cartesian graph Programme F4: Graphs Follow these instructions to plot the graph of:

18. STROUD Worked examples and exercises are in the text Using a spreadsheet Construction of a Cartesian graph Programme F4: Graphs 1.Enter the number –1 in A1 2.Highlight the cells A1 to A12 3.Select Edit-Fill-Series and in the Series window change the Step value from 1 to 0.3 and Click OK

19. STROUD Worked examples and exercises are in the text Using a spreadsheet Construction of a Cartesian graph Programme F4: Graphs 4.Enter the formula =(A1-2)^3 in B1 5.Activate B1 and select Edit-Copy 6.Highlight B2 to B12 and select Edit-Paste 7.Highlight the cells A1:B12 8.Click the Chart Wizard button

20. STROUD Worked examples and exercises are in the text Using a spreadsheet Construction of a Cartesian graph Programme F4: Graphs 9.Click XY (Scatter)

21. STROUD Worked examples and exercises are in the text Using a spreadsheet Construction of a Cartesian graph Programme F4: Graphs 10.Click top right-hand corner type 11.Click Next

22. STROUD Worked examples and exercises are in the text Using a spreadsheet Construction of a Cartesian graph Programme F4: Graphs 12.Click Legend tab 13.Clear the tick 14.Click the Titles tab 15.Enter in the Value (X) Axis x-axis 16.Enter in the Value (Y) Axis y-axis 17.Click Next

23. STROUD Worked examples and exercises are in the text Using a spreadsheet Construction of a Cartesian graph Programme F4: Graphs 18.Ensure the lower radio button is selected 19.Click Finish

24. STROUD Worked examples and exercises are in the text Using a spreadsheet Construction of a Cartesian graph Programme F4: Graphs The graph of y = (x – 2) 3

25. STROUD Worked examples and exercises are in the text Graphs of equations Using a spreadsheet Inequalities Absolute values Programme F4: Graphs

26. STROUD Worked examples and exercises are in the text Inequalities Less than or greater than Programme F4: Graphs The inequality y > x states that whatever value is chosen for the independent variable x the corresponding value of the dependent variable y is greater. There is an infinity of values of y greater than any finite chosen value of x so the plot produces an area rather than a line.

27. STROUD Worked examples and exercises are in the text Graphs of equations Using a spreadsheet Inequalities Absolute values Programme F4: Graphs

28. STROUD Worked examples and exercises are in the text Absolute values Modulus Graphs Inequalities Interaction Programme F4: Graphs

29. STROUD Worked examples and exercises are in the text Absolute values Modulus Programme F4: Graphs When numbers are plotted on a straight line the distance a given number from zero is called the absolute value or modulus of that number. For example, the absolute value of –5 is 5 because it is 5 units distant from 0 and the absolute value of 3 is 3 because it is 3 units distant from 3.

30. STROUD Worked examples and exercises are in the text Absolute values Graphs Programme F4: Graphs Using a spreadsheet to plot the graph of y = |x| the built-in function ABS is used. 1.Fill cells A1 to A21 with numbers in the range –5 to 5 (step 0.5) 2.In cell B1 type the formula =ABS(A1) 3.Copy the contents of B1 into B2 – B21

31. STROUD Worked examples and exercises are in the text Absolute values Graphs Programme F4: Graphs 4.Highlight cells A1:B21 and draw the graph of y = |x|.

32. STROUD Worked examples and exercises are in the text Absolute values Inequalities Programme F4: Graphs A line drawn parallel to the x-axis though the point y = 2 intersects the graph at x = ±2. So that if y < 2, that is |x| < 2 then –2 < x < 2 and if y > 2, that is |x| > 2 then x 2.

33. STROUD Worked examples and exercises are in the text Absolute values Inequalities Programme F4: Graphs In general if: |x − a| < b then –b < x – a < b so that a – b < x < a + b and if: |x − a| > b then x – a b so that x a + b

34. STROUD Worked examples and exercises are in the text Absolute values Interaction Programme F4: Graphs The spreadsheet can be used to demonstrate dynamically how changing features of an equation affect the appearance of the graph.

35. STROUD Worked examples and exercises are in the text Programme F4: Graphs Learning outcomes Construct a collection of ordered pairs of numbers from an equation Plot points associated with ordered pairs of numbers against Cartesian axes and generate graphs Appreciate the existence of asymptotes to curves and discontinuities Use a spreadsheet to draw Cartesian graphs of equations Describe regions of the x–y plane that are represented by inequalities