f(x)→ g(x): Vertical translation by 2 units down
f(x)→ h(x): Horizontal translation by 2 units to the right
Practice makes perfect
We want to use the function f to graph g and h. We also want to describe the transformations from the graph of f to the graphs of g and h. We will begin by graphing f(x)=3x+1. To do it, let's use a table of values to find points on the graph of f.
x
3x+1
f(x)
-2
3( -2)+1
-5
0
3( 0)+1
1
2
3( 2)+1
7
We can plot these points and connect them with a straight line to have the graph of f(x).
Function g(x)
Now, let's look at how the function g(x)=f(x)-2 differs from f(x). We will make a table of values again. This time to graph g. We will use the same x -values so that we can compare them.
x
f(x)
f(x)-2
g(x)
-2
-5
-5-2
-7
0
1
1-2
-1
2
7
7-2
5
If we plot these points on the same coordinate plane as f(x), we can compare the two functions.
We see that each y-value is being translated 2 units down. This is a vertical translation.
Function h(x)
We can go through the same process with h(x)=f(x-2).
x
f(x)
f(x-2)
h(x)
-2
-5
3( -2-2)+1
-11
0
1
3( 0-2)+1
-5
2
7
3( 2-2)+1
1
If we plot these points on the same coordinate plane as f(x), we can compare the two functions.
We see that each y-value is being translated 2 units to the right. This is a horizontal translation.
