Associated Topics || Dr. Math Home || Search Dr. Math

### Translating Functions

```
Date: 08/27/98 at 18:29:57
From: Geoff Mosley
Subject: Functions

Find f(2+h), f(x+h), and f(x+h)-f(x)/h where h cannot = 0 for
f(x) = x/(x + 1)?

Explain how the following graphs are obtained from the graph of
y = f(x):

y = f(x - 5)
y = -f(x)
y = f(5x)
```

```
Date: 08/28/98 at 12:31:12
From: Doctor Peterson
Subject: Re: Functions

Hi, Geoff.

I get the impression you are at the beginning of a calculus course.

For the first group of problems, all you have to do is substitute
(2+h), or (x+h), for x in the definition of the function, and then
simplify if you can. For example, f(2+h) is:

(2 + h)
-----------
(2 + h) + 1

and you can do just a little simplifying.

The second group contains nice things to know. Here's how to think
about it. Imagine you have just graphed a point of f(x), say at x = k:

|
|       /
+      *
|     /
|
----+------+-------
|      k

Now you want to graph a point of f(x - 5) using what you just found
out. Well, you know f(k). If x = k + 5, f(x-5) = f(k) which you just
figured out. So the corresponding point in the new function to plot is
x = k + 5, where it will have the same value:

|
|       /    /
+      *    o
|     /    /
|
----+------+----+--
|      k   k+5

If you do this with every point in the graph, you will find that the
graph of f(x - 5) is just the graph of f(x) slid to the right by 5.
The other two cases are similar. Find what point of the new graph feeds
the same value into f as the original, and what y is for the new
function.

You may have seen this before if you had an equation like:

y = k*(x-a)^2 + b

and had to find the vertex of the parabola. Notice that this is just:

y = f(x-a) + b

where:

f(x) = k*x^2

whose vertex is at (0,0). So the vertex of the original equation is at
(a,b). Knowing how transforming an equation transforms its graph is
very useful.

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Equations, Graphs, Translations
High School Functions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search