Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2016 | June | Q#4

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Question Figure  shows a sketch of part of the curve with equation y = f(x). The curve has a maximum point A  at (–2, 4) and a minimum point B at (3, –8) and passes through the origin O.

On separate diagrams, sketch the curve with equation

a.   y = 3f(x),

b.   y = f(x) – 4

On each diagram, show clearly the coordinates of the maximum and the minimum points and the  coordinates of the point where the curve crosses the y-axis.

Solution

a.

We are given the sketch of the curve with equation; We are required to sketch the curve of equation; We know that and represent ‘stretched’ 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 vertical expansion of the given function.

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

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

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 three-times of original given function.

It is shown in the figure below. b.

We are given graph of y=f(x).

We are required to sketch y=f(x)-4.

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

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

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

 Original Transformed Translation Vector Movement Function     units in negative y-direction Coordinates  It is evident that y=f(x)-4 is a case of translation by 4 units along negative y-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.

To sketch y=f(x) – 4, we simply shift this y=f(x) graph 4 units along negative y-axis as shown below. 