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
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.
Drawing Other Graphs
Often the easiest way to draw a graph is to construct a table of values.
Draw y = x² + 3x + 2 for -3 £ x £ 3
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
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.
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
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.
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
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
2) Make the equations you need to solve like the one you have the
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
Copyright © Matthew Pinkney 2003