Question

a.   Factorise completely 9x – 4x³

b.   Sketch the curve C with equation

y = 9x – 4x³

Show on your sketch the coordinates at which the curve meets the x-axis.

The points A and B lie on C and have x coordinates of –2 and 1 respectively.

c.   Show that the length of AB is  where k is a constant to be found.

Solution

a.
 

We are given;

We have algebraic formula;

Therefore;

b.    

We are required to sketch;

It is evident that it is a cubic equation.

We can now sketch the curve as follows.

ü Find the sign of the coefficient of . This gives the shape of the graph at the extremities. 

It is evident that with negative coefficient of  will shape the curve at extremities like decreasing  from left to right.

ü Find the point where the graph crosses y-axis by finding the value of  when .

We can find the coordinates of y-intercept from the given equation of the curve. 

Hence, the curve crosses y-axis at point .

ü Find the point(s) where the graph crosses the x-axis by finding the value of  when . If  there is repeated root the graph will touch the x-axis.

We can find the coordinates of x-intercepts from the given equation of the curve.

Now we have two options.

Hence, the curve crosses x-axis at points  and .

ü Calculate the values of  for some value of . This is particularly useful in determining the  quadrant in which the graph might turn close to the y-axis.

ü Complete the sketch of the graph by joining the sections.

ü Sketch should show the main features of the graph and also, where possible, values where the graph intersects coordinate axes.

c.    

We are given that the points A and B lie on C and have x coordinates of –2 and 1 respectively.

We need to find the length of AB.

Expression for the distance between two given points  and is:

Therefore we need coordinates of the points A and B.

If a point lies on the curve (or the line), the coordinates of that point satisfy the equation of the curve  (or the line).

Therefore, we substitute x-coordinates of both points A and B in given equation of the curve to find  the corresponding y-coordinates.

For point A, substitute x=-2;	For point A, substitute x=1;

Hence the points A and B have coordinates  and , respectively. 

Now we can find the distance AB.

Since ;