http://www.ck12.org Chapter 1. Functions and Graphs
Look carefully at the vertex of the parabola. It has moved to the right two units. This is because first the entire graph
was shifted entirely to the right 6 units. Then the function was horizontally compressed by a factor of 3 which means
the point (6, 0) became (2, 0) and thexvalue of every other point was also compressed by a factor of 3 towards the
linex=0. This method is counter-intuitive because it requires reading the transformations backwards (the opposite
of the way the order of operations tells you to).
Alternatively, the argument can be factored and each component of the transformation will present itself. This time
the stretch occurs from the center of the transformed graph, not the origin. This method is ultimately the preferred
method.
f( 3 (x− 2 ))
Either way, this is a horizontal compression by a factor of 3 and a horizontal shift to the right by 2 units.
Example B
Describe the transformation in words and show an example with a picture:f(x)→−^14 f(x)+ 3
Solution:This is a vertical stretch by a factor of^14 , a reflection over thexaxis, and a vertical shift 3 units up. As
opposed to what you saw in Example A, the order of the transformations for anything outside of the argument is
directly what the order of operations dictates.
First, the parabola is reflected over thex axis and compressed vertically so it appears wider. Then, every
point is moved up 3 units.
Example C
Describe the transformation in words and show an example with a picture:
f(x)→− 3 f(−^12 x− 1 )+ 1
Solution:Every possible transformation is occurring in this example. The horizontal and the vertical components
do not interact with each other and so your description of the transformation can begin with either component. Here,
start by describing the vertical components of the transformation:
There is reflection across the x axis and a vertical stretch by a factor of 3. Then, there is a vertical shift up 1 unit.
Below is an image of a non-specific function going through the vertical transformations.