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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.

We can also sometimes use the graph of one equation to solve another.

#### Example

Draw the graph of y = x² - 3x + 5 .
Use this graph to solve 3x + 1 - x² = 0 and x² - 3x - 6 = 0

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, the left hand side is our y above, so to solve the equation, we find the values of x when y = 6 (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.