Now, let's look at all of the possible transformations so that we can more clearly identify what is happening to our function.
Transformations of f(x)
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k>0
y=f(x)- k
Horizontal Translations
Translation right h units, h>0
y=f(x- h)
Translation left h units, h>0
y=f(x+ h)
Vertical Stretch or Compression
Vertical stretch, a>1
y= af(x)
Vertical compression, 0< a< 1
y= af(x)
Horizontal Stretch or Compression
Horizontal stretch, 0< b<1
y=f( bx)
Horizontal compression, b>1
y=f( bx)
Reflections
In the x-axis
y=- f(x)
In the y-axis
y=f(- x)
As we can see, we can describe our transformation as a vertical translation 2 units down. It means that the graph of the given function was shifted 2 units down.
b Using the function f(x), we want to find the expression - 2 * f(x). To do this, we will multiply the given function by - 2 and simplify.
We can describe these transformations as a reflection in the x-axis and a vertical stretch by a factor of 2.
c Using the function f(x) we want to evaluate for the given value, f( x-2). To do this, we need to substitute x-2 for x in each instance of the x-variable.
We can describe this transformation as a horizontal translation 2 units right. It means that the graph of the given function was shifted 2 units right.
d Using the function f(x), we want to evaluate for the given value, f( - 2x). To do this, we need to substitute - 2x for x in each instance of the x-variable and simplify.
