Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | January | Q#10

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Question

a.   On the axes below, sketch the graphs of


i.       


ii.    

showing clearly the coordinates of all the points where the curves cross the coordinate axes.

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

Solution

a.
 


i.
 

We are required to sketch;

We need to expand it in order to sketch it.

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 right to left.

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.

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 three 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.

desmos-graph.png


ii.
 

We are required to sketch;

If we first sketch , it would become easy to sketch .

A graph of the form  is known as a reciprocal graph and looks like;

desmos-graph (2).png

Next we sketch;

Transformation of the function  into  results from ‘stretch’ of  in y- direction by a scale factor of .

‘Stretch’ of the function  in y-direction by a scale factor of  transforms  into  .

Original

Transformed

Effect

Function

Expansion
Vertically by

Coordinates

Function

Shrinking
Horizontally by

Coordinates

Function

Shrinking
Vertically by

Coordinates

Function

Expansion
Horizontally by

Coordinates

It is evident that  is a case of vertical expansion by a scale factor of 2.

desmos-graph (3).png

Now we are ready to sketch;

Transformation of the function  into  results from reflection of  in x-axis. 

Reflection of the function  in x-axis transforms  into .

Original

Transformed

Reflection
in

Function

x-axis

Coordinates

It is evident that we are required to transform the function  into ,  therefore it is  case of reflection of the given function in x-axis.

It is also evident from the above table that only y-coordinates of the graph change whereas x- coordinates of the graph will remain unchanged. 

Hence, the new function has all the x-coordinates same as that of original given function whereas  all the y-coordinates are negative of original given function.

It is shown in the figure below. 

desmos-graph (2).png

b.
 

We are given that;

It is evident that this equation is obtained by equating the equations of the two given graphs in (a). 

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).

Therefore, the equation given in (b) gives the point(s) of intersection of the two given graphs in (a).

We can sketch the both graphs sketched in (a) on a single graph as shown below.

desmos-graph (8).png

Since the two graphs intersect each other at two distinct points, therefore, there are will be only two  solutions of the given equation.

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