Question

Figure 1 shows a sketch of the curve C with equation y = f(x) where

f (x) = x²(9 –2x)

There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the x-axis at the point  A.

a.   Write down the coordinates of the point A.

b.   On separate diagrams sketch the curve with equation

                i.       y = f(x + 3)

                  ii.       y = f(3x)

On each sketch you should indicate clearly the coordinates of the maximum point and any points  where the curves cross or meet the coordinate axes.

The curve with equation y = f (x) + k, where k is a constant, has a maximum point at (3, 10).

c.   Write down the value of k.

Solution

a.
 

We are given that the curve C cuts x-axis at point A, hence, A is x-intercept of the given curve. 

We are required to find coordinates of the point A.

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).

We are given that equation of the curve C is;

We substitute y=0 in this equation to find the coordinates of x-intercept.

Now we have two options.

Hence, the curve C cuts x-axis at two points one where x=0 and other where  . Since there is  minimum point of the curve at origin, x=0, belongs to this point and  belongs to point A.

Hence, coordinates of point A are .

b.    

                                    i.
 

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 3 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.

                                  ii.
 

We are given the sketch of the curve with equation;

We are required to sketch the curve of equation;

We know that  and  represent ‘stretch’ 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 horizontal shrinking of the given  function.

Transformation of the function  into  results from shrinking of  in x- direction by a scale factor of .

Shrinking of the function  in x-direction by a scale factor of  transforms  into  .

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 the original given function whereas all the x-coordinates are one-third of the original given function.

It is shown in the figure below. 

c.
 

We are given that the curve with equation y = f (x) + k, where k is a constant, has a maximum point  at (3, 10) and we are required to find the value of k.

We are given graph of y=f(x).

We are required to sketch y=f(x)+k.

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)+k is a case of translation by k units along positive y-axis. 

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.

To sketch y=f(x)+k, we simply shift this y=f(x) graph k units along positive y-axis as. However, we  are given that maximum point of the curve shifts from (3,27) to (3,10) which represents  translation by 17 units along negative y-axis.

Therefore, k=-17.