Graph Transformations
There are many times when you’ll know very well what the graph of a
particular function looks like, and you’ll want to know what the graph of a
very similar function looks like. In this chapter, we’ll discuss some ways to
draw graphs in these circumstances.
Transformations “after” the original function
Suppose you know what the graph of a function f(x) looks like. Suppose
d 2 R is some number that is greater than 0, and you are asked to graph the
function f(x) + d. The graph of the new function is easy to describe: just
take every point in the graph of f(x), and move it up a distance of d. That
is, if (a, b) is a point in the graph of f(x), then (a, b + d) is a point in the
graph of f(x) + d.
As an explanation for what’s written above: If (a, b) is a point in the graph
of f(x), then that means f(a) = b. Hence, f(a) + d = b + d, which is to say
that (a, b + d) is a point in the graph of f(x) + d.
The chart on the next page describes how to use the graph of f(x) to create
the graph of some similar functions. Throughout the chart, d > 0, c > 1, and
(a, b) is a point in the graph of f(x).
Notice that all of the “new functions” in the chart di↵er from f(x) by some
algebraic manipulation that happens after f plays its part as a function. For
example, first you put x into the function, then f(x) is what comes out. The
function has done its job. Only after f has done its job do you add d to get
the new function f(x) + d. 67Because all of the algebraic transformations occur after the function does
its job, all of the changes to points in the second column of the chart occur
in the second coordinate. Thus, all the changes in the graphs occur in the
vertical measurements of the graph.
New How points in graph of f(x) visual e↵ect
function become points of new graph
f(x) + d (a, b) 7! (a, b + d) shift up by d
f(x) Transformations before and after the original function
As long as there is only one type of operation involved “inside the function”
– either multiplication or addition – and only one type of operation involved
“outside of the function” – either multiplication or addition – you can apply
the rules from the two charts on page 68 and 70 to transform the graph of a
function.
Examples.
• Let’s look at the function • The graph of 2g(3x) is obtained from the graph of g(x) by shrinking
the horizontal coordinate by 1
3, and stretching the vertical coordinate by 2.
(You’d get the same answer here if you reversed the order of the transfor-
mations and stretched vertically by 2 before shrinking horizontally by 1
3. The
order isn’t important.)
74
7:—
(x) 4,
7c’
‘I
II
‘I’
-I
is the component for state 
, then you have
. So the state vector is 
.
