Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2016 | June | Q#4
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;
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Original |
Transformed |
Effect |
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Function |
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Expansion |
Coordinates |
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Function |
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Shrinking |
Coordinates |
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Function |
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Shrinking |
Coordinates |
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Function |
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Expansion |
Coordinates |
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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 |
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Function |
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Coordinates |
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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.
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