Transformations of functions:
Dilatations.
4th year of secondary education. (Group B).
 
Vertical dilatation. 

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 with the form y = af(x), where a is any real number.

You can increase or decrease the value of parameter a 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 multiplied by constant a is in red. Change the parameter a to values between one and zero and then to different values of more than one and watch what happens.

Note that when 0<a<1, the graph reduces vertically until a*100%. However when a>1, the graph increases vertically until it reaches a*100% of its initial size.

Vertical dilatation of a function times a is equivalent to multiplying all the y-coordinates of the graph of the function by a. It is the same as multiplying the scale of the OY axis by a. Do the following exercise in your exercise book to check this is true:

Exercise 1: Draw a set of axes in your exercise book making the scale of OY axis double the scale of the OX axis. Now copy the graph of the function f(x)=x3-3x onto the set of axes. Check that it is the same as the graph of the function g(x)=2(x3-3x) in the window above. (Just change parameter a to 2)

Exercise 2: Look carefully at what happens to the local maximum and minimum points of f(x)=x3-3x when a=2. If the function f(x) has a minimum point at (2,-3) where is the corresponding local minimum point of the function 5*f(x)?

Horizontal dilatation.

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 with the form y = f(ax), 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 equation of the function

g(x)=(ax)3-3(ax)

is in red. Change the parameter a to values between one and zero and then to different values of more than one and watch what happens.

As you will have noticed when 0<a<1, the graph increases horizontally until (1/a)*100%. However when a>1, the graph reduces horizontally until it reaches (1/a)*100%.

Horizontal dilatation of a function times a is equivalent to multiplying all the x-coordinates of the graph of the function by 1/a. It is the same as multiplying the scale of the OX axis by 1/a. Do the following exercise in your exercise book to check this is true:

Exercise 1: Draw a set of axes in your exercise book making the scale of OX axis double the scale of the OY axis. Now copy the graph of the function f(x)=x3-3x onto the set of axes. Check that it is the same as the graph of the function g(x)=(x/2)3-3(x/2) in the window above. (Just change parameter a to 1/2)

Exercise 2: Look carefully at what happens to the local maximum and minimum points of f(x)=x3-3x when a=2 and when a=0.2. If the function f(x) has a maximum point at (1,-2) where is the corresponding maximum point of the function f(0.25x)?

Exercise 3: If the function f(x) cuts the OX axis at (4,0) where does the function f(4x) cut the OX axis?

Changing Scale.

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 with the form y = af(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 equation of the function

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

is in red. Change the parameter a to values between one and zero and then to different values of more than one and watch what happens.

Note that when 0<a<1, the graph reduces proportionally until a*100%. However when a>1, the graph increases proportionally until a*100%.

It is equivalent to multiplying both the x and y coordinates of the graph of the function by a. This is the same as multiplying the scale by a. Do the following exercise to check this is true:

Exercise 1: Change parameter a to 2 in the window above. The graph of the function f(x)=x3-3x is in blue and the graph of the function g(x)=2((x/2)3-3(x/2)) is in red. We are increasing the graph of f(x) to up to 200%, i.e. double. Now make the scale double and check that the graph that was red then becomes the blue graph. Reduce the scale and double it again until you can see the change clearly. This therefore shows that the graph of the function g(x) is the same as that of f(x) when the scale is doubled.

Exercise 2: Look carefully at what happens to the local maximum and minimum points of f(x)=x3-3x when a=3. If the function f(x) has a minimum point at (-1,2) where is the corresponding minimum point of the function 5f(x/5)?

Exercise 3:Use the Descartes window above to draw the graph of the following function: (Use the space on the right where the function is written in blue).

f(x)=sin(x)

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

f(x)=2sinx f(x)=sin(x/2) f(x)=2sin(x/2)

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

       
           
  Francisco José Merayo González
 
Spanish Ministry of Education. Year 2001
 
 

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