# 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. 