STROUD Worked examples and exercises are in the text PROGRAMME F4 (from 6 th Ed) GRAPHS (revised 29 Jan 14 – J.A.B.)

STROUD

Worked examples and exercises are in the text

PROGRAMME F4(from 6th Ed)

GRAPHS

(revised 29 Jan 14 – J.A.B.)

STROUD


Graphs of equations

(Using a spreadsheet)

Inequalities

Absolute values

Programme F4: Graphs

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Graphs of equations

Using a spreadsheet

Inequalities

Absolute values


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Graphs of equations

Equations

Ordered pairs of numbers

Cartesian axes

Drawing a graph


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Graphs of equations

Equations


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).

A conditional equation is a statement of the equality of two expressions that is only true for restricted values of the symbols involved.

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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.

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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.

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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.

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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.

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Equation of a straight line

Copied from 6th Ed Prog 8

(but in 7th Ed Prog 13)

The basic equation of a straight line is:where:

y mx c

gradient

intercept on the -axis

dym

dxc y

When m is negative, the line slopes downwards. (ADDED by JAB)

m is also called the slope. Can be found from the coordinates of any two points (x1,y1) and (x2,y2) . We get m = (y2 - y1) / (x2 - x1).

(The dy/dx thing above is a “derivative” and will be covered later.)

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Graphs of equations

Using a spreadsheet (mostly removed – study in textbook if you wish)

Inequalities

Absolute values


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Processing numbers

The graph of y = x3

copied from Programme F10: Functions

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Using a spreadsheet

Construction of a Cartesian graph


The graph of y = (x – 2)3

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Processing numbers

Graphs of inverses


The ordered pairs of input-output numbers that are used to generate the graph of a function are reversed for the inverse function.

Consequently, the graph of the inverse of a function is the shape of the graph of the original function reflected in the line y = x.

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Processing numbers

The graph of y = x1/3


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Processing numbers

The graphs of y = x3 and y = x1/3 plotted together


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Variants of quadratic functions and cubic functions [ on board in class – J.A.B.]

I.e., various cases of:

y = ax2 + bx + c [quadratic]

y = ax3 + bx2 + cx + d [cubic]

Documents

STROUD Worked examples and exercises are in the text PROGRAMME F4 (from 6 th Ed) GRAPHS (revised 29 Jan 14 – J.A.B.)