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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Graphs

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Contents 1 The Equation of a Straight Line 1.1 Example 2 Graphs of Quadratic Equations 2.1 Examples 3 Drawing Other Graphs 3.1 Example 4 Intersecting Graphs 5 Simultaneous Equations 5.1 Example 6 Solving Equations 6.1 Example 7 Comments

The Equation of a Straight Line

Equations of straight lines are in the form

( and are numbers).

is the gradient of the line and is the y-intercept (where the graph crosses the y-axis).

NB1: If you are given the equation of a straight-line and there is a number before the 'y', divide everything by this number to get y by itself, so that you can see what and are.

NB2: Parallel lines have equal gradients.

Example

(diagram of the graph needed)

The above graph has equation:

which is the same as ).

Gradient = change in y / change in x .

It cuts the y-axis at -2, and this is the constant () in the equation.

Note: for all straight line graphs, is always the gradient of the line and is always the point the line crosses the y-axis.

Graphs of Quadratic Equations

These are curves and will have a turning point. Remember, quadratic equations are of the form:

(, and are numbers).

If is positive, the graph will be 'U' shaped. If is negative, the graph will be 'n' shaped. The graph will always cross the y-axis at the point (so is the y-intercept point). Graphs of quadratic functions are sometimes known as parabolas.

Examples

(graphs for an examples needed)

Drawing Other Graphs

Often the easiest way to draw a graph is to construct a table of values and then plot the points.

Example

Draw: for .

x	-3	-2	-1	0	1	2	3
x^2	9	4	1	0	1	4	9
3x	-9	-6	-3	0	3	6	9
2	2	2	2	2	2	2	2
y	2	0	0	2	6	12	20

The table shows that when , , and . Since , we add up the three values in the table to find out what is when , etc.

We then plot the values of x and y on graph paper and draw a smooth curve through all the points (WE DO NOT JOIN THEM UP 'DOT-TO-DOT' WITH STRAIGHT LINES).

(diagram of the plotted curve needed)

Intersecting Graphs

If we wish to know the coordinates of the point(s) where two graphs intersect, we solve the equations simultaneously. This can be done using the graphs.

Simultaneous Equations

You can solve simultaneous equations by drawing graphs of the two equations you wish to solve. The x and y values of where the graphs intersect are the solutions to the equations.

Example

Solve the simultaneous equations:

and by graphical methods.

To do this, you simply plot the two equations on the same set of axes - as shown below. Their solution is where they cross.

(graph for an example needed)

From the graph, and (approx.). These are the answers to the simultaneous equations.

Solving Equations

Any equation can be solved by drawing a graph of the equation in question. The points where the graph crosses the x-axis are the solutions. So if you asked to solve:

using a graph, draw the graph of:

and the points where the graph crosses the x-axis are the solutions to the equation.

Example

If you are asked to draw the graph of:

and then are asked to use this graph to solve:

and

,

you would proceed in the following way:

1. Make a table of values for and draw the graph.
2. Make the equations you need to solve like the one you have the graph of. So for :
   1. Multiply both sides by -1 to get
   2. Add 6 to both sides: .

Now we know that and , therefore, . Find out what is when and these are the answers (you should get two answers).

Try solving yourself using your graph of . You should get a answers of around -1.4 and 4.4 .

Comments

The simultaneous section should have the example written and then linked to the simultaneous article stating there are other methods to solve them. The graphical section of the simultaneous article could link here.