Question

Figure 1 shows a sketch of a curve with equation y = f(x).

The curve crosses the y-axis at (0, 3) and has a minimum at P (4, 2).

On separate diagrams, sketch the curve with equation

a.   y = f(x + 4),

b.    y = 2f(x).

On each diagram show the coordinates of minimum point and any point of intersection with the y- axis.

Solution

a.
 

We are given the sketch of the curve with equation;

We are required to sketch the curve of equation;

Translation through vector  transforms the graph of  into the graph of . 

Transformation of the function  into  results from translation through vector  .

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

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

However, for the given case we consider following.

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

Translation through vector  transforms the function  into . 

Transformation of the function  into  results from translation through vector  .

Translation through vector  transforms the function  into  which means shift towards left along x-axis. 

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

It is evident that we are required to transform the function  into , therefore it is  case of translation of  along negative x-axis by 4 unit.

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

Hence, the new function has all the y-coordinates same as that of original given function whereas  all the x-coordinates are shifted towards negative x-axis of original given function. 

It is shown in the figure below.

b.
 

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 two-times of original given function.

It is shown in the figure below.