Transformations of functions:
Translations.
4th year of secondary education (Group B).
 
Vertical translation.

In this section we are going to analyse how we can use the graph of the function y = f(x) to help us to easily draw the graph of any function which has the form y = f(x) + b, where b is any real number.

You can increase or decrease the value of the parameter b by using the arrows at the bottom of the window.

In this window we can see the graph of the function f(x)=x3-3x. There are two equations underneath: the equation of this function is written in blue and the equation of the same function where the constant b has been added is in red. Change the value of b giving it both positive and negative values and watch what happens.

Note that when b>0 the graph moves b units vertically upwards and when b<0 it moves vertically downwards. In other words, we are translating the vector v(0,b) in such a way that the y-coordinates for all the points on the graph either increase or decrease by b units.

Note that the corresponding local maximum and minimum point of the function also move b units vertically when we change the value of parameter b.

Exercise: If f(x) has a local maximum at the point (2,1) where would the corresponding maximum point be for the function f(x)-5?

Horizontal translation. 

In this section we are going to analyse how we can use the graph of the function y = f(x) to help us to easily draw the graph of any function which has the form y = f(x-a), where a is any real number.

In this window we can see the graph of the function f(x)=x3-3x. There are two equations underneath: the equation of this function is written in blue and the following function in red

g(x)=(x-a)3-3(x-a).

Change the value of the parameter a, giving it both positive and negative values, and watch what happens.

Note that when a>0 the graph moves a units horizontally towards the right and when a<0 it moves horizontally to the left. In other words, we are translating the vector v(a,0) in such a way that the x-coordinates for all the points on the graph either increase or decrease by a units.

Note that the corresponding local maximum and minimum points of the function also move a units horizontally when we change the value of parameter a.

Exercise: If f(x) has a local minimum value at the point (-1,4), where would the corresponding local minimum point be for the function f(x+3)?

Oblique translation.

In this section we are going to analyse how we can use the graph of the function y = f(x) to help us to easily draw the graph of any function which has the form  y = f(x-a) + b, where a and b are real numbers. From what we have already seen it is obvious that two translations will take place, a vertical and a horizontal one. 

In this window we can see the graph of the function f(x)=x4-2x2. There are two equations underneath: this equation is written in blue and the following equation in red:

g(x)=(x-a)4-2(x-a)2+b.

Change the values of parameters a and b, giving them both positive and negative values, and watch what happens.

Note that the graph moves vertically and horizontally according to the values of a and b. In other words, we are translating the vector v(a,b) in such a way that we obtain the coordinates of all the points on the graph f(x-a)+b by adding (a,b) to them.

Exercise 1: If f(x) has a local maximum point at (-2,3) where is the corresponding local maximum point for the function f(x+1)-5?

Exercise 2: Use the Descartes window above to draw the graph of the following functions: (Use the space on the right where you can see the function in blue).

Then, copy these same graphs into your exercise book on graph paper. Use these graphs and the steps outlined on this page to help you to draw graphs of these functions:

Finally, check that your work is correct by changing the functions in the window above or by keeping the original functions and altering the parameters a and b appropriately in the same window.

       
           
  Francisco José Merayo González
 
Spanish Ministry of Education. Year 2001
 
 

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