Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2017  June  Q#9
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Question
a. On separate axes sketch the graphs of
i. y = –3x + c, where c is a positive constant,
ii.
On each sketch show the coordinates of any point at which the graph crosses the
yaxis and the equation of any horizontal asymptote.
Given that y = –3x + c, where c is a positive constant, meets the curve
b. show that (5 – c)^{2} >12
c. Hence find the range of possible values for c.
Solution
a.
We are required to sketch a line and a curve.
i.
First we sketch the line.
We are given equation of the line as;
First we find the xintercept of the line.
The point
Therefore, we substitute
Hence, coordinates of xintercept of the line are
Next, we find the yintercept of the line.
The point
Therefore, we substitute
Hence, coordinates of yintercept of the line are
We can sketch the line as shown below.
ii.
We are required to sketch the graph and given that;
It may be a lot easier to first sketch;
A graph of the form
We have the sketch of the curve with equation
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 positive yaxis.
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 symptote 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 (orange line) can be stated with equation;
Similarly, vertical asymptote (blue line) is yaxis and can be stated with equation;
b.
We are given that y = –3x + c, where c is a positive constant, meets the curve
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;
Two values of x indicate that there are two intersection points.
Since we are given that the line and the curve intersect at two distinct points, this quadratic equation must have two distinct roots.
For a quadratic equation
Where
If
If
If
Therefore, for this quadratic equation;
c.
We are required to solve the inequality;
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