Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2013  January  Q#6
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Question
Figure 1 shows a sketch of the curve with equation , x ≠ 0.
The curve C has equation , x ≠ 0, and the line
a. Sketch and clearly label the graphs of C and
On your diagram, show clearly the coordinates of the points where C and
b. Write down the equations of the asymptotes of the curve C.
c. Find the coordinates of the points of intersection of
Solution
a.
We are required to sketch the graphs of curve C and line
Let us first sketch he curve C.
We are given equation of the curve C as;
However, we are also given graph of the curve with equation;
It is evident that curve C is a transformed version of the given curve whose sketch is already given.
We are given that;
We are required to sketch y=f(x)5.
Translation through vector
Translation through vector
Transformation of the function
Translation vector
Original 
Transformed 
Translation Vector 
Movement 

Function 




Coordinates 


It is evident that y=f(x)5 is a case of translation by 5 units along negative yaxis.
We just need to translate the given sketch of the curve along negative yaxis by 5 units as shown below. The red is the original while orange is the translated graph.
Next we need to sketch the line
A line can be sketch easily by marking its x and y intercepts.
The point
Therefore, we substitute y=0 in equation of the line;
Hence coordinates of yintercept are
The point
Therefore, we substitute x=0 in equation of the line;
Hence coordinates of yintercept are (0,2).
Therefore, the line
b.
We are given that;
Next we are required to state the equations of asymptotes.
An asymptote is a line that a curve approaches, as it heads towards infinity. Both horizontal and vertical asymptotes may exist for a given graph. The distance between the curve and the asymptote tends to zero as they head to infinity.
For the graph sketched above, we can draw horizontal and vertical asymptotes as shown below.
It is evident that horizontal asymptote (blue line) can be stated with equation;
Similarly, vertical asymptote (green line) can be stated with equation;
c.
We are required to find the coordinates of point(s) of intersection of curve C and line
If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate
Equation of the line is;
Equation of the curve is;
Equating both equations;
Now we have two options.









Two values of x indicate that there are two intersection points.
With xcoordinate of point of intersection of two lines (or line and the curve) at hand, we can find the ycoordinate of the point of intersection of two lines (or line and the curve) by substituting value of xcoordinate of the point of intersection in any of the two equations.
We choose equation of the line
For 
For 






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