Question

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

 ,

The curve crosses the x-axis at (1, 0), touches it at (–3, 0) and crosses the y-axis at (0, –9).

a.   In the space below, sketch the curve C with equation y=f(x+2) and state the coordinates of the  points where the curve C meets the x-axis.

b.   Write down an equation of the curve C.

c.   Use your answer to part (b) to find the coordinates of the point where the curve C meets the y- axis.

Solution

a.
 

We are given sketch of the curve  and are required to sketch the graph of the curve  .

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 2 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, red one is the original while orange is the transformed graph.

b.
 

We are given;

We are can write the equation of the curve C by replacing x with x+2;

c.
 

We are required to find the coordinates of the point where the curve  C meets the y-axis. We are looking for y-intercept of the curve C.

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).

We have found in (b) equation of the curve C;

Substitute x=0;

Hence, coordinates of y-intercept of the curve C are (0,25).