PROGRAMME F4

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Worked examples and exercises are in the text

PROGRAMME F4

GRAPHS

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Graphs of equationsUsing a spreadsheetInequalitiesAbsolute values

Programme F4: Graphs

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Graphs of equationsEquations

Ordered pairs of numbers

Cartesian axes

Drawing a graph


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Graphs of equationsEquations


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).

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Graphs of equationsOrdered 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.

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Graphs of equationsCartesian 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.

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Graphs of equationsDrawing 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.

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Graphs of equationsDrawing 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.

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Using a spreadsheetSpreadsheets

Rows and columns

Text and number entry

Formulas

Clearing entries

Construction of a Cartesian graph


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Using a spreadsheetSpreadsheets


Electronic spreadsheets provide extensive graphing capabilities and their use is widespread. All descriptions here are based on the Microsoft spreadsheet Excel 97 for Windows.

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Using a spreadsheetRows 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.

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Using a spreadsheetText and number entry


Every cell on the spreadsheet is capable of having numbers or text entered into it via the keyboard.

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Using a spreadsheetFormulas


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).

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Using a spreadsheetClearing 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.

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Using a spreadsheetConstruction of a Cartesian graph


Follow these instructions to plot the graph of:

3( 2)y x

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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

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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

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9. Click XY (Scatter)

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10. Click top right-hand corner type

11. Click Next

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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

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18. Ensure the lower radio button is selected

19. Click Finish

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The graph of y = (x – 2)3

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InequalitiesLess 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.

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Absolute valuesModulus

Graphs

Inequalities

Interaction


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Absolute valuesModulus


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.

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Absolute valuesGraphs


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

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Absolute valuesGraphs


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

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Absolute valuesInequalities


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.

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Absolute valuesInequalities


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

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Absolute valuesInteraction


The spreadsheet can be used to demonstrate dynamically how changing features of an equation affect the appearance of the graph.

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Programme F4: GraphsLearning 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

PROGRAMME F4

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Transcript of PROGRAMME F4