Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | January | Q#5

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Question

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

 ,

The curve passes through the origin and has two asymptotes, with equations y=1 and x=2, as  shown in Figure.

a.   In the space below, sketch the curve with equation y = f (x −1) and state the equations of the  asymptotes of this curve.

b.   Find the coordinates of the points where the curve with equation y = f (x −1) crosses the  coordinate axes.

Solution

a.
 

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 positive x-axis by 1 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 positive x-axis of original given function.

Same would be effect on equations of asymptotes. The x=2 will translate to x=3 whereas y=1 will  remain unchanged.

It is shown in the figure below.

b.
 

We are given the sketch of the curve with equation;

 ,

The equation for  is obtained by replacing  by ;

Therefore equation will be;

As demonstrated in (a), the horizontal translation of y=f(x) to y=f(x-1), along positive x-axis by unit 1,  will translate x-intercept of the curve from origin (0,0) to (1,0).

We need to find y-intercept.

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

Therefore;

Hence, coordinates of y-intercept are

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