The Teaching Tutors - The Transformations of Graphs of Functions is Easy!

Reference & EducationEducation

  • Author Kimberly Lauziere
  • Published May 24, 2010
  • Word count 671

The graph of any function may be transformed either by shifting, stretching/compressing, or reflection.

For shifting and stretching/compressing, there are two types: horizontal and vertical.

A graph may also be reflected either over the x-axis or the y-axis.

  1. Vertical transformations:

A shift may also be referred to as a translation. In order to vertically translate the graph of y = f(x) by c units upward, c has to be added to the function. The function now becomes y = f(x) + c.

For a downward translation of c units, the function becomes y = f(x) – c. Note that in this case, c is subtracted from the function y = f(x).

In general, a vertical translation means that every point (x, y) on the graph of y = f(x) is transformed to (x, y + c) on the graph of y = f(x) + c. On the other hand, every point (x, y) on the graph of y = f(x) is transformed to (x, y – c) on the graph of y = f(x) – c.

  1. Horizontal transformations:

Horizontal translations of the function y = f(x) are dealt with in a different manner.

When the function is shifted c units to the right, x becomes (x – c) so that the new function is y = f(x – c). When the same function y = f(x) is translated c units to the left, the new function becomes y = f(x + c).

Another way of looking at this is to remember that a horizontal translation means that every point (x, y) on the graph of y = f(x) is transformed to (x + c, y) on the graph of y = f(x – c). On the other hand, every point (x, y) on the graph of y = f(x) is transformed to (x – c, y) on the graph of y = f(x + c).

  1. Vertical Stretching and Compression:

The next type of transformation is vertical stretching and compression.

If y = f(x) represents the graph of the original function as mentioned above, then a graph affected by vertical stretching or compression is expressed as y = cf(x). It should be noted that when 0 < c < 1, a vertical shrinking of the graph of y = f(x) is observed. Graphically, a vertical shrinking "pulls" the graph of y = f(x) toward the x-axis.

When c > 1 in the function y = cf(x), the graph is "pushed" away from the x-axis (vertical stretching). The x-intercept remains the same as the original function in both cases.

Another way to think of vertical shrinking and compressing is that every point (x, y) on the graph of y = f(x) is transformed to (x, cy) on the graph of y = cf(x).

  1. Horizontal Stretching and Compression:

Next, one has to analyze horizontal stretching and compression.

If y = f(x) represents the graph of the original function as mentioned above, then a graph affected by horizontal stretching or compression is expressed as y = f(cx). It should be noted that when 0 < c < 1, a horizontal stretching of the graph of y = f(x) is observed. Graphically, a horizontal stretching "pulls" the graph of y = f(x) away from the y-axis.

When c > 1 in the function y = f(cx), the graph is "pulled" toward the y-axis (horizontal shrinking). The y-intercept remains the same as the original function in both cases.

In addition, horizontal shrinking and compressing means that every point (x, y) on the graph of y = f(x) is transformed to (x/c, y) on the graph of y = f(cx).

  1. Reflection:

The last type of transformation to look at is the reflection. It is fairly straightforward to understand!

When the original graph of y = f(x) is reflected across the x-axis, the function of the reflected graph becomes y = –f(x). On the other hand, when the same function is reflected across the y-axis, the function of the reflected graph is y = f(–x). That’s it!

Remember that the transformations mentioned above may be combined within the same function so that one graph can be shifted, stretched, and reflected!

For over ten years, we have provided Private Tutoring Services enrolled in home schools and traditional schools and helped them achieve their academic goals as well as outstanding grades in mathematics, English, science, literature, and language courses.If You need Home Tutoring Services Please visit our website:http://theteachingtutors.com/

Article source: https://articlebiz.com
This article has been viewed 750 times.

Rate article

Article comments

There are no posted comments.

Related articles