Influence of a transformation of the equation of a function on the graph




f(x) and f(x-a)

We compare the functions with equation y = f(x) and y = f(x-3). To realize what this means, we can take an example:
 
y  = 3 x2 + 4 x + 5    and

y  = 3 (x-3)2 + 4 (x-3) + 5
The two graphs are parabolas.

The image of a by the first function is

3. a2 + 4.a + 5

If we want the same image for the second function, we have to replace x by a+3. Then we have the image

3. ((a+3)-3)2 + 4.((a+3)-3) + 5 = 3. a2 + 4.a + 5

Generalizing: the image of x by the first function is the same as the image of x+3 by the second function.

If we write the first function as f(x), then the second function is f(x-3). This means that the graph of y=f(x-3) arises when we move the graph of y = f(x) three units to the right.

Exercise : plot the two functions and notice the translation.
The graph of y=f(x-a) arises when we move the graph of y = f(x) exactly a units to the right.

f(x) and f(x) + a

We compare the functions with equation y = f(x) and f(x) + a.

For all x, we see that the image of f(x) + a is exactly a units more than the image of f(x).

The graph of y=f(x) + a arises when we move the graph of y = f(x) exactly a units upwards.

Translation of the graph of y=f(x) by a vector v

We choose a arbitrary fixed vector v(a,b).
We shift the graph of y=f(x) by the vector v. First we move the graph a units to the right. Then we move the graph b units upwards. The new equation is y = f(x-a) + b

Conclusion:
Is we submit the graph of y= f(x) to a translation by the vector v(a,b), then the image is the function with equation y = f(x-a) + b.

f(x) and f(a.x)

We compare the functions with equation y = f(x) and y = f(3x).

To realize what this means, we can take an example:

 
y  =  x2 - x   en

y  = (3x)2 - (3x)
The image of a by the first function is

a2 - a

If we want the same image for the second function, we have to start with x = a/3. Then we have the image

(3. a/3 )2 - (3. a/3) = a2 - a

Generalizing: the image of x by the first function is the same as the image of x/3 by the second function.

This means that the graph of y=f(3x) arises when we compress the graph of y = f(x) towards the y-axis with a factor 3.

The graph of y=f(ax) arises when we compress the graph of y = f(x) towards the y-axis with a factor a.

Note: compress the graph of y = f(x) towards the y-axis with a factor 1/4 is the same as stretching the graph away from the y-axis with factor 4.

Exercise: plot y = x2 and y = (x/4)2.

Composing transformations

Example 1

We start with the function with equation y = sin(x).

Example 2

We start with the parabola with equation y = 3(x-1)2 - 4

Example 3

We start with the function with equation y = x2 -x

Example 4

We start with the function with equation y = 5 cos(2x-6) + 4
We want a suitable composition of elementary transformations such that the equation simplifies to y = cos(x).

Example 5

It is also possible to build up the graph of y = 5 cos(2x-6)+ 4 starting from the graph of y = cos(x).

Solved Problems

These following exercises assume the knowledge of the trigonometry chapter.


Topics and Problems

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