Question

Figure 1 shows a sketch of the curve C with equation

 , x ≠ 0.

The curve C crosses the x-axis at the point A.

a.   State the x coordinate of the point A.

The curve D has equation y = x²(x – 2), for all real values of x.

b.   A copy of Figure 1 is shown below.

On this copy, sketch a graph of curve D.

Show on the sketch the coordinates of each point where the curve D crosses the coordinate axes.

Sketch and clearly label the graphs of C and  on a single diagram.

c.   Using your sketch, state, giving a reason, the number of real solutions to the equation

Solution

a.
 

We are given the curve C with equation

We are required to find the coordinates of x-intercept of the curve;

The point  at which curve (or line) intercepts x-axis, the value of . So we can find the  value of  coordinate by substituting  in the equation of the curve (or line).

We substitute y=0 in given equation of the curve;

Hence, coordinates of the x-intercept of the curve are (-1,0).

 

b.
 

We are required to sketch;

We are given;

We can rewrite it as;

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 points  and .

It can be seen that  has come twice therefore the curve touches the x-axis at this point.

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 an expression which is equating both equations (of the curves C and D).

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e.  coordinates of that point have same values on both lines (or on the line and the curve).  Therefore, we can equate  coordinates of both lines i.e. equate equations of both the lines (or the  line and the curve).

Hence, solution of this equation will yield x-coordinates of points of intersection of both curves C  and D.

Two values of x indicate that there are two intersection points.

It can be seen from (b) that both curves intersect at two points, therefore, this equation will also  have two solutions.