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

y-axis and the equation of any horizontal asymptote.

Given that y = –3x + c, where c is a positive constant, meets the curve  at two distinct  points,

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 x-intercept of the line.

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).

Therefore, we substitute  in equation of the line.

Hence, coordinates of x-intercept of the line are .

Next, we find the y-intercept of the line.

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).

Therefore, we substitute  in equation of the line.

Hence, coordinates of y-intercept 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  is known as a reciprocal graph and looks like;

We have the sketch of the curve with equation . 

We are required to sketch y=f(x)+5.

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
Function				units in positive y-direction
Coordinates

It is evident that y=f(x)+5 is a case of translation by 5 units along positive y-axis.

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 y-axis and can be stated with equation;

b.
 

We are given that y = –3x + c, where c is a positive constant, meets the curve  at two  distinct points, therefore, the line and the curve intersect at two distinct point.

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  coordinates of both lines i.e. equate equations of both the lines (or the  line and the curve).

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 , the expression for solution is;

Where  is called discriminant.

If , the equation will have two distinct roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

Therefore, for this quadratic equation;

c.    

We are required to solve the inequality;