Cambridge O Level Additional Mathematics 4037
You must know the chapters co-ordinate geometry and logarithm before starting this chapter.
Equation of straight line graphs:
The equation of a straight line is
y = mx + c
where, m is the gradient;
c is the y - intercept.
The formula of gradient, m = (y2 - y1)/(x2 - x1)
In these sums, you are given an equation which you have to transform into a straight line equation. The type of equations that you get are: y = ax^n and y = Ab^x .
You have to transform these to straight line equations.
We use logarithmic functions to transform these.
How to change y=Ab^x to a straight line.
y = A * b^x
log y = log (A* b^x)
log y = log A + log b^x
log y = log A + x log b
log y = (log b)x + log A
If log y is plotted against x, the graph would be a straight line with gradient as (log b) and y-intercept (log A).
How to change y=ax^n to a straight line.
y = a * x^n
log y = log (a * x^n)
log y = log a + log x^n
log y = log a + n log x
log y = n log x + log a
If log y is plotted against log x, a straight line is obtained with gradient n and y-intercept (log a).
These equations can also be replaced with ln.
Next you have to draw the straight line. It may be a graph of log y against x or log y against log x (depending on the equation of your question). Plot and draw this graph as perfectly as you can.
Log can also be replaced by ln if the range of ln seems more suitable.
There are usually unknown quantities in your equation which you have to find out from the graph that you have drawn (e.g. gradient, intercept).
For example, in the equation y=Ab^x, the gradient is log b. Therefore, if the gradient that you got from your graph is 15,
then log b = 1.5
log 10^ b = 1.5
therefore, b = 10^1.5
And this is pretty much it. There might be slight variations in the questioning pattern but your main concept remains the same. You get an equation which you have to change into a straight line equation and draw the line to find the unknown constants in your equation.
Lets solve a question now.
This is an extra note just in case you need clarification.
Identifying gradient and y-intercept from an equation:
I am going to give a very simple example.
The first equation is the equation of a straight line and you have to find out the gradient and the y-intercept from the second equation.
As you can see below, the gradient is the coefficient of x and the remaining part is the y-intercept.
Hence,
m = 2p^3;
c = 4q^2/5n
You may sometimes have to rearrange the equation to the form y = mx + c to identify m and c.
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