Question

A curve with equation y=f(x) passes through the point (3,6). Given that

a.   use integration to find f(x). Give your answer as a polynomial in its simplest form.

b.   Show that , where p is a positive constant. State the value of p.

c.   Sketch the graph of y = f(x), showing the coordinates of any points where the curve touches or  crosses the coordinate axes.

Solution

a.
 

We are given;

We are given coordinates of a point on the curve (3,6).

We are required to find the equation of y in terms of x ie f(x).

We can find equation of the curve from its derivative through integration;

Therefore,

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

If a point   lies on the curve , we can find out value of . We substitute values of  and    in the equation obtained from integration of the derivative of the curve i.e. .

Therefore, substituting the coordinates of point (3,6) in above equation;

Therefore, equation of the curve C is;

b.
 

We have found in (b) that;

We are given that;

We can now compare the given and the found equations.

This yields that;

Hence, p=3 and we can write the given expression as;

c.    

We are required to sketch;

As demonstrated in (b), we can write it as;

Substituting p=3 as found in (b);

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 positive coefficient of  will shape the curve at extremities like increasing 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 two 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.