Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2012 | June | Q#10

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Question

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

f (x) = x2(9 –2x)

There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point  A.

a.   Write down the coordinates of the point A.

b.   On separate diagrams sketch the curve with equation

                i.       y = f(x + 3)

                  ii.       y = f(3x)

On each sketch you should indicate clearly the coordinates of the maximum point and any points  where the curves cross or meet the coordinate axes.

The curve with equation y = f (x) + k, where k is a constant, has a maximum point at (3, 10).

c.   Write down the value of k.

Solution

a.
 

We are given that the curve C cuts x-axis at point A, hence, A is x-intercept of the given curve. 

We are required to find coordinates of the point A.

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 are given that equation of the curve C is;

We substitute y=0 in this equation to find the coordinates of x-intercept.

Now we have two options.

Hence, the curve C cuts x-axis at two points one where x=0 and other where  . Since there is  minimum point of the curve at origin, x=0, belongs to this point and  belongs to point A.

Hence, coordinates of point A are .

b.    

                                    i.
 

We are given the sketch of the curve with equation;

We are required to sketch the curve of equation;

Translation through vector  transforms the graph of  into the graph of

Transformation of the function  into  results from translation through vector  .

Translation through vector  represents the move,  units in the positive x-direction and  units in
the positive y-direction.

Original

Transformed

Translation

Vector

Movement

Function

 units in
positive x-direction

 units in
positive y-direction

Coordinates

However, for the given case we consider following.

Translation through vector  represents the move,  units in the negative x-direction and  units  in the y-direction.

Translation through vector  transforms the function  into

Transformation of the function  into  results from translation through vector  .

Translation through vector  transforms the function  into  which means shift towards left along x-axis. 

Original

Transformed

Translation

Vector

Movement

Function

 units in
negative x-direction

Coordinates

It is evident that we are required to transform the function  into , therefore it is  case of translation of  along negative x-axis by 3 unit.

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 original given function whereas  all the x-coordinates are shifted towards negative x-axis of original given function. 

It is shown in the figure below.

                                  ii.
 

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

Effect

Function

Expansion
Vertically by

Coordinates

Function

Shrinking
Horizontally by

Coordinates

Function

Shrinking
Vertically 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 shrinking of the given  function.

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

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

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 are one-third of the original given function.

It is shown in the figure below. 

c.
 

We are given that the curve with equation y = f (x) + k, where k is a constant, has a maximum point  at (3, 10) and we are required to find the value of k.

We are given graph of y=f(x).

We are required to sketch y=f(x)+k.

Translation through vector  represents the move,  units in the x-direction and  units in the  positive y-direction.

Translation through vector  transforms the function  into  or

Transformation of the function  into  or  results from translation  through vector .

Translation vector  transforms the function  into  or  which  means shift upwards along y-axis.

Original

Transformed

Translation

Vector

Movement

Function

 units in
positive y-direction

Coordinates

It is evident that y=f(x)+k is a case of translation by k units along positive y-axis. 

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.

To sketch y=f(x)+k, we simply shift this y=f(x) graph k units along positive y-axis as. However, we  are given that maximum point of the curve shifts from (3,27) to (3,10) which represents  translation by 17 units along negative y-axis.

Therefore, k=-17.

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