Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01R) | Year 2013 | June | Q#9

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Question

Figure 1 shows a sketch of the curve C with equation y = f(x).

The curve C passes through the point (-1,0) and touches the x-axis at the point (2,0).

The curve C has a maximum at the point (0,4).

a.   The equation of the curve C can be written in the form

Where a, b and c are integers.

Calculate the values of a, b and c.

b.   Sketch the curve with equation  in the space provided on page 24.

Show clearly the coordinates of all the points where the curve crosses or meets the coordinate  axes.

Solution

a.

We are given the sketch of the curve C with equation y = f(x). It is evident that it is a cubic graph  and it is evident from the given general equation of the curve, as well.

If  is a polynomial of degree  then  will have exactly  factors, some of which may repeat.

A cubic equation can be written as a product of its all three factors as follows;

This is case where all three roots of the cubic polynomial are distinct. This means the cubic graph  crosses the x-axis at three distinct points with coordinates  and

However, if cubic graph has a repeated roots and one distinct roots then;

In this case, one factor  is repeated and other  is distinct. This means the cubic graph  crosses the x-axis at two distinct points with coordinates  and . The repeated root shows  that the cubic graph only touches the x-axis at this point .

From the given sketch of the curve C we can see that graph crosses x-axis at point (-1,0) and  touches it at point (2,0).
Hence, factors of the cubic graph are
and repeated .

Therefore;

We can expand the right hand side, to equate with the left hand side.

Comparing both sides yields that;

b.

We are given the sketch of the curve with equation;

We are required to sketch the curve of equation;

We know that  and  represent
‘stretch’ in transformation of given functions. Here
, therefore;

 Original Transformed Value of Effect Function – ‘Stretch’ Horizontally by Coordinates Function Shrinking Horizontally by Coordinates Function Expansion Horizontally by Coordinates Function Shrinking Horizontally by Coordinates Function Expansion Horizontally by Coordinates

From the above table, as highlighted, it is evident that we are required to transform the function   into , where , therefore it is case of horizontal expansion of the given  function.

Transformation of the function  into  results from expansion of  in x- direction by a scale factor of  if .

Expansion of the function  in x-direction by a scale factor of  transforms  into   if .

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

Hence, the new function has all the y-coordinates same as that of the original given function  whereas all the x-coordinates will be ‘c’ times the original given function.

It is shown in the figure below.

The red graph is the original one whereas orange graph represents the transformed one.

It is evident that only x-coordinates of the graph are  times the original ones whereas y-coordinates  remain unchanged.