Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663A/01) | Year 2014 | January | Q#4
Question
Figure 1 shows a sketch of a curve with equation y = f(x).
The curve crosses the y-axis at (0, 3) and has a minimum at P (4, 2).
On separate diagrams, sketch the curve with equation
a. y = f(x + 4),
b. y = 2f(x).
On each diagram show the coordinates of minimum point and any point of intersection with the y- axis.
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 |
|
|
|
|
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 |
|
|
|
|
Coordinates |
|
|
It is evident that we are required to transform the function into
, therefore it is case of translation of
along negative x-axis by 4 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 negative x-axis 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 ‘stretched’ in transformation of given functions. Here
, therefore;
|
|||
Original |
Transformed |
Effect |
|
Function |
|
|
Expansion |
Coordinates |
|
|
|
Function |
|
|
Shrinking |
Coordinates |
|
|
|
Function |
|
|
Shrinking |
Coordinates |
|
|
|
Function |
|
|
Expansion |
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 two-times of original given function.
It is shown in the figure below.
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