STROUD Worked examples and exercises are in the text 1 STROUD Worked examples and exercises are in the text Programme F4: Graphs PROGRAMME F4 GRAPHS

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

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

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

STROUD Worked examples and exercises are in the text 5 STROUD Worked examples and exercises are in the text Programme F4: Graphs Graphs of equations Equations 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).

STROUD Worked examples and exercises are in the text 6 STROUD Worked examples and exercises are in the text Programme F4: Graphs Graphs of equations Ordered pairs of numbers 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.

STROUD Worked examples and exercises are in the text 7 STROUD Worked examples and exercises are in the text Programme F4: Graphs Graphs of equations Cartesian axes 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.

STROUD Worked examples and exercises are in the text 8 STROUD Worked examples and exercises are in the text Programme F4: Graphs Graphs of equations Drawing a graph 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.

STROUD Worked examples and exercises are in the text 9 STROUD Worked examples and exercises are in the text Programme F4: Graphs Graphs of equations Drawing a graph 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.

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

STROUD Worked examples and exercises are in the text 11 STROUD Worked examples and exercises are in the text Programme F4: Graphs Graphs of equations Using a spreadsheet Inequalities Absolute values

STROUD Worked examples and exercises are in the text 12 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Spreadsheets Rows and columns Text and number entry Formulas Clearing entries Construction of a Cartesian graph

STROUD Worked examples and exercises are in the text 13 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Spreadsheets Electronic spreadsheets provide extensive graphing capabilities and their use is widespread. All descriptions here are based on the Microsoft spreadsheet Excel 2010.

STROUD Worked examples and exercises are in the text 14 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Rows and columns 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.

STROUD Worked examples and exercises are in the text 15 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Text and number entry Every cell on the spreadsheet is capable of having numbers or text entered into it via the keyboard.

STROUD Worked examples and exercises are in the text 16 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Formulas 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).

STROUD Worked examples and exercises are in the text 17 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Clearing entries 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.

STROUD Worked examples and exercises are in the text 18 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Construction of a Cartesian graph Follow these instructions to plot the graph of:

STROUD Worked examples and exercises are in the text 19 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Construction of a Cartesian graph 1.Enter the number –1 in A1 2.Highlight the cells A1 to A15 3.Select Edit-Fill-Series and in the Series window change the Step value from 1 to 0.3 and Click OK

STROUD Worked examples and exercises are in the text 20 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Construction of a Cartesian graph 4.Enter the formula =(A1-2)^3 in B1 5.Activate B1 and select Edit-Copy 6.Highlight B2 to B15 and select Edit-Paste 7.Highlight the cells A1:B15 8.Click the Charts tab followed by Scatter and then Smooth Lines Scatter

STROUD Worked examples and exercises are in the text 21 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Construction of a Cartesian graph 9. Click Chart Layout tab 10. Select Legend then No Legend 11. Click the Axis Titles tab 12. Select Horizontal and Below Type in x-axis 13. Enter Vertical and Horizontal Type in y-axis

STROUD Worked examples and exercises are in the text 22 STROUD Worked examples and exercises are in the text Programme F4: Graphs Using a spreadsheet Construction of a Cartesian graph The graph of y = (x – 2) 3

STROUD Worked examples and exercises are in the text 23 STROUD Worked examples and exercises are in the text Programme F4: Graphs Graphs of equations Using a spreadsheet Inequalities Absolute values

STROUD Worked examples and exercises are in the text 24 STROUD Worked examples and exercises are in the text Programme F4: Graphs Graphs of equations Using a spreadsheet Inequalities Absolute values

STROUD Worked examples and exercises are in the text 25 STROUD Worked examples and exercises are in the text Programme F4: Graphs Inequalities Less than or greater than 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.

STROUD Worked examples and exercises are in the text 26 STROUD Worked examples and exercises are in the text Programme F4: Graphs Graphs of equations Using a spreadsheet Inequalities Absolute values

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

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

STROUD Worked examples and exercises are in the text 29 STROUD Worked examples and exercises are in the text Programme F4: Graphs Absolute values Modulus 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.

STROUD Worked examples and exercises are in the text 30 STROUD Worked examples and exercises are in the text Programme F4: Graphs Absolute values 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

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

STROUD Worked examples and exercises are in the text 32 STROUD Worked examples and exercises are in the text Programme F4: Graphs Absolute values Inequalities 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.

STROUD Worked examples and exercises are in the text 33 STROUD Worked examples and exercises are in the text Programme F4: Graphs Absolute values Inequalities 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

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

STROUD Worked examples and exercises are in the text 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