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PROGRAMME F4. GRAPHS. Programme F4: Graphs. Graphs of equations Using a spreadsheet Inequalities Absolute values. Programme F4: Graphs. Graphs of equations Using a spreadsheet Inequalities Absolute values. Programme F4: Graphs. Graphs of equations Equations

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slide2

Programme F4: Graphs

Graphs of equations

Using a spreadsheet

Inequalities

Absolute values

slide3

Programme F4: Graphs

Graphs of equations

Using a spreadsheet

Inequalities

Absolute values

slide4

Programme F4: Graphs

Graphs of equations

Equations

Ordered pairs of numbers

Cartesian axes

Drawing a graph

slide5

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).

slide6

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.

slide7

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.

slide8

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.

slide9

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.

slide10

Programme F4: Graphs

Graphs of equations

Using a spreadsheet

Inequalities

Absolute values

slide11

Programme F4: Graphs

Using a spreadsheet

Spreadsheets

Rows and columns

Text and number entry

Formulas

Clearing entries

Construction of a Cartesian graph

slide12

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 97 for Windows.

slide13

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.

slide14

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.

slide15

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).

slide16

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.

slide17

Programme F4: Graphs

Using a spreadsheet

Construction of a Cartesian graph

Follow these instructions to plot the graph of:

slide18

Programme F4: Graphs

Using a spreadsheet

Construction of a Cartesian graph

  • Enter the number –1 in A1
  • Highlight the cells A1 to A12
  • Select Edit-Fill-Series and in the Series window change the Step value from 1 to 0.3 and Click OK
slide19

Programme F4: Graphs

Using a spreadsheet

Construction of a Cartesian graph

  • Enter the formula =(A1-2)^3 in B1
  • Activate B1 and select Edit-Copy
  • Highlight B2 to B12 and select Edit-Paste
  • Highlight the cells A1:B12
  • Click the Chart Wizard button
slide20

Programme F4: Graphs

Using a spreadsheet

Construction of a Cartesian graph

  • Click XY (Scatter)
slide21

Programme F4: Graphs

Using a spreadsheet

Construction of a Cartesian graph

  • Click top right-hand corner type
  • Click Next
slide22

Programme F4: Graphs

Using a spreadsheet

Construction of a Cartesian graph

  • Click Legend tab
  • Clear the tick
  • Click the Titles tab
  • Enter in the Value (X) Axis
  • x-axis
  • Enter in the Value (Y) Axis
  • y-axis
  • Click Next
slide23

Programme F4: Graphs

Using a spreadsheet

Construction of a Cartesian graph

  • Ensure the lower radio button is selected
  • Click Finish
slide24

Programme F4: Graphs

Using a spreadsheet

Construction of a Cartesian graph

The graph of y = (x – 2)3

slide25

Programme F4: Graphs

Graphs of equations

Using a spreadsheet

Inequalities

Absolute values

slide26

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.

slide27

Programme F4: Graphs

Graphs of equations

Using a spreadsheet

Inequalities

Absolute values

slide28

Programme F4: Graphs

Absolute values

Modulus

Graphs

Inequalities

Interaction

slide29

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.

slide30

Programme F4: Graphs

Absolute values

Graphs

  • Using a spreadsheet to plot the graph of y = |x| the built-in function ABS is used.
  • Fill cells A1 to A21 with numbers in the range –5 to 5 (step 0.5)
  • In cell B1 type the formula =ABS(A1)
  • Copy the contents of B1 into B2 – B21
slide31

Programme F4: Graphs

Absolute values

Graphs

  • Highlight cells A1:B21 and draw the graph of y = |x|.
slide32

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 or x > 2.

slide33

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 or x – a> b so that

x < a – b or x > a + b

slide34

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.

slide35

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
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