# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663A/01) | Year 2014 | January | Q#9

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. 