Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2013 | June | Q#8
Question
Figure 1 shows a sketch of the curve with equation y = f(x) where
,
The curve crosses the x-axis at (1, 0), touches it at (–3, 0) and crosses the y-axis at (0, –9).
a. In the space below, sketch the curve C with equation y=f(x+2) and state the coordinates of the points where the curve C meets the x-axis.
b. Write down an equation of the curve C.
c. Use your answer to part (b) to find the coordinates of the point where the curve C meets the y- axis.
Solution
a.
We are given sketch of the curve and are required to sketch the graph of the curve
.
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 2 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, red one is the original while orange is the transformed graph.
b.
We are given;
We are can write the equation of the curve C by replacing x with x+2;
c.
We are required to find the coordinates of the point where the curve C meets the y-axis. We are looking for y-intercept of the curve C.
The point at which curve (or line) intercepts y-axis, the value of
. So we can find the value of
coordinate by substituting
in the equation of the curve (or line).
We have found in (b) equation of the curve C;
Substitute x=0;
Hence, coordinates of y-intercept of the curve C are (0,25).
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