Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2012 | June | Q#10
Question
Figure 1 shows a sketch of the curve C with equation y = f(x) where
f (x) = x2(9 –2x)
There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point A.
a. Write down the coordinates of the point A.
b. On separate diagrams sketch the curve with equation
i. y = f(x + 3)
ii. y = f(3x)
On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes.
The curve with equation y = f (x) + k, where k is a constant, has a maximum point at (3, 10).
c. Write down the value of k.
Solution
a.
We are given that the curve C cuts x-axis at point A, hence, A is x-intercept of the given curve.
We are required to find coordinates of the point A.
The point at which curve (or line) intercepts x-axis, the value of
. So we can find the value of
coordinate by substituting
in the equation of the curve (or line).
We are given that equation of the curve C is;
We substitute y=0 in this equation to find the coordinates of x-intercept.
Now we have two options.
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Hence, the curve C cuts x-axis at two points one where x=0 and other where . Since there is minimum point of the curve at origin, x=0, belongs to this point and
belongs to point A.
Hence, coordinates of point A are .
b.
i.
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 |
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Function |
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Coordinates |
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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 |
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Function |
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Coordinates |
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It is evident that we are required to transform the function into
, therefore it is case of translation of
along negative x-axis by 3 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.
ii.
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
, 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 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 one-third of the original given function.
It is shown in the figure below.
c.
We are given that the curve with equation y = f (x) + k, where k is a constant, has a maximum point at (3, 10) and we are required to find the value of k.
We are given graph of y=f(x).
We are required to sketch y=f(x)+k.
Translation through vector represents the move,
units in the x-direction and
units in the positive y-direction.
Translation through vector transforms the function
into
or
.
Transformation of the function into
or
results from translation through vector
.
Translation vector transforms the function
into
or
which means shift upwards 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)+k is a case of translation by k units along positive y-axis.
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.
To sketch y=f(x)+k, we simply shift this y=f(x) graph k units along positive y-axis as. However, we are given that maximum point of the curve shifts from (3,27) to (3,10) which represents translation by 17 units along negative y-axis.
Therefore, k=-17.
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