Question

Figure 1 shows a sketch of the curve with equation  , x ≠ 0.

The curve C has equation , x ≠ 0, and the line  has equation y = 4x + 2.

a.   Sketch and clearly label the graphs of C and  on a single diagram.

On your diagram, show clearly the coordinates of the points where C and  cross the coordinate  axes.

b.   Write down the equations of the asymptotes of the curve C.

c.   Find the coordinates of the points of intersection of  and y = 4x + 2.

Solution

a.
 

We are required to sketch the graphs of curve C and line  on the same diagram.

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  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 negative y-direction
Coordinates

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

We just need to translate the given sketch of the curve along negative y-axis by 5 units as shown  below. The red is the original while orange is the translated graph.

Next we need to sketch the line  for which equation is;

A line can be sketch easily by marking its x and y intercepts.

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 y=0 in equation of the line;

Hence coordinates of y-intercept are .

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 x=0 in equation of the line;

Hence coordinates of y-intercept are (0,2).

Therefore, the line  can be sketched on the same diagram as shown below.

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  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;

Now we have two options.

Two values of x indicate that there are two intersection points.

With x-coordinate of point of intersection of two lines (or line and the curve) at hand, we can find the  y-coordinate of the point of intersection of two lines (or line and the curve) by substituting value  of x-coordinate of the point of intersection in any of the two equations.

We choose equation of the line ;

For	For