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
yintercept (where the graph crosses the yaxis).
NB1: If you are given the equation of a straightline 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 yaxis 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 yaxis at the point c (so c is the yintercept 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 xaxis 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 xaxis 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
Answer:
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.
Copyright © Matthew Pinkney 2003
