Graphs

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 .