Question

Figure  shows a sketch of part of the curve with equation y = f(x). The curve has a maximum point A  at (–2, 4) and a minimum point B at (3, –8) and passes through the origin O. 

On separate diagrams, sketch the curve with equation

a.   y = 3f(x),

b.   y = f(x) – 4

On each diagram, show clearly the coordinates of the maximum and the minimum points and the  coordinates of the point where the curve crosses the y-axis.

Solution

a.
 

We are given the sketch of the curve with equation;

We are required to sketch the curve of equation;

We know that  and  represent ‘stretched’ in transformation of given functions.  Here , therefore;

From the above table, as highlighted, it is evident that we are required to transform the function  into , where , therefore it is case of vertical expansion of the given function.

Transformation of the function  into  results from expansion of  in y- direction by a scale factor of  if .

Expansion of the function  in y-direction by a scale factor of  transforms  into   if .

It is also evident from the above table that only y-coordinates of the graph change whereas x- coordinates of the graph will remain unchanged. 

Hence, the new function has all the x-coordinates same as that of original given function whereas  all the y-coordinates are three-times of original given function. 

It is shown in the figure below.

b.
 

We are given graph of y=f(x).

We are required to sketch y=f(x)-4.

Translation through vector  represents the move,  units in the x-direction and   units in the negative y-direction.

Translation through vector  transforms the function  into  or . 

Transformation of the function  into  or  results from translation  through vector .

Translation through vector  transforms the function  into  or  which means shift downwards along y-axis.

	Original	Transformed	Translation Vector	Movement
Function				units in negative y-direction
Coordinates

It is evident that y=f(x)-4 is a case of translation by 4 units along negative y-axis. 

It is also evident from the above table that only y-coordinates of the graph change whereas x- coordinates of the graph will remain unchanged.

To sketch y=f(x) – 4, we simply shift this y=f(x) graph 4 units along negative y-axis as shown below.