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

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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 Graphs of equations (Using a spreadsheet) Inequalities Absolute values Programme F4: Graphs

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

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

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

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.

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.

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.

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.

STROUD Worked examples and exercises are in the text Equation of a straight line Copied from 6 th Ed Prog 8 (but in 7 th Ed Prog 13) The basic equation of a straight line is: where: 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 (x 1,y 1 ) and (x 2,y 2 ). We get m = (y 2 - y 1 ) / (x 2 - x 1 ). (The dy/dx thing above is a “derivative” and will be covered later.)

STROUD Worked examples and exercises are in the text Graphs of equations Using a spreadsheet (mostly removed – study in textbook if you wish) Inequalities Absolute values Programme F4: Graphs

STROUD Worked examples and exercises are in the text Processing numbers The graph of y = x 3 copied from Programme F10: Functions

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

STROUD Worked examples and exercises are in the text Processing numbers Graphs of inverses copied from Programme F10: Functions 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.

STROUD Worked examples and exercises are in the text Processing numbers The graph of y = x 1/3 copied from Programme F10: Functions

STROUD Worked examples and exercises are in the text Processing numbers The graphs of y = x 3 and y = x 1/3 plotted together copied from Programme F10: Functions

STROUD Worked examples and exercises are in the text Variants of quadratic functions and cubic functions [ on board in class – J.A.B.] I.e., various cases of: y = ax 2 + bx + c [quadratic] y = ax 3 + bx 2 + cx + d [cubic]

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