Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2005 | January | Q#6

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Question

Figure 1 shows a sketch of the curve with equation y = f(x). The curve crosses the x-axis at the  points (2, 0) and (4, 0). The minimum point on the curve is P(3, –2).

In separate diagrams sketch the curve with equation

a.   y=–f(x),

b.   y=f(2x).

On each diagram, give the coordinates of the points at which the curve crosses the x-axis, and the  coordinates of the image of P under the given transformation.

Solution

a.
 

We are given the sketch of the curve with equation;

We are required to sketch the curve of equation;

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.

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 .

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 half of the original given function.

It is shown in the figure below.

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