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**The Equation of a Straight Line**

Equations of straight lines are in the form y = mx + c (m and c are numbers). m is the gradient of the line and c 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 m and c are.

NB2: Parallel lines have equal gradients.

The above graph has equation y = (4/3)x – 2 (which is the same as 3y + 6 = 4x).

Gradient = change in y / change in x = 4 / 3

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

**Graphs of Quadratic Equations**

These are curves and will have a turning point. Remember, quadratic equations are of the form: y = ax² + bx + c (a, b and c are numbers). If ‘a’ is positive, the graph will be ‘U’ shaped. If ‘a’ is negative, the graph will be ‘n’ shaped. The graph will always cross the y-axis at the point c (so c is the y-intercept point). Graphs of quadratic functions are sometimes known as parabolas.

*Example*:

**Drawing Other Graphs**

Often the easiest way to draw a graph is to construct a table of values.

*Example*:

Draw y = x² + 3x + 2 for -3 £ x £ 3

__x -3 -2 -1 0 1 2 3__

x² 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 x = -3, x² = 9, 3x = -9 and 2 = 2. Since y = x² + 3x + 2, we add up the three values in the table to find out what y is when x = -3, etc.

We then plot the values of x and y on graph paper.

**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 3y = -2x + 6 and y = 2x -2 by graphical methods.

From the graph, y = 1 and x = 1.5 (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 x² – 3 = 0 using a graph, draw the graph of y = x² – 3 and the points where the graph crosses the x-axis are the solutions to the equation.

If you are asked to draw the graph of y = x² – 3x + 5 and then are asked to use this graph to solve 3x + 1 – x² = 0 and x² – 3x – 6 = 0, you would proceed in the following way:

1) Make a table of values for y = x² – 3x + 5 and draw the graph.

2) Make the equations you need to solve like the one you have the graph of. So for 3x + 1 – x² = 0:

i) multiply both sides by -1 to get x² – 3x -1 = 0

ii) add 6 to both sides: x² – 3x + 5 = 6

Now we know that y = x² – 3x + 5 and x² – 3x + 5 = 6, therefore, y = 6. Find out what x is when y = 6 and these are the answers (you should get two answers).

Try solving x² – 3x – 6 = 0 yourself using your graph of y = x² – 3x + 5. You should get a answers of around -1.4 and 4.4 .