Plot both of the graphs and then compare some points on the graphs.
Translates the graph 3 units to the left and 2 units down
Practice makes perfect
We are given a rational function f(x)= 7x. We will find the needed translation to get the function y= 7x+3-2. To do so we will graph both of them. Let's start by recalling the general equation of a rational function.
y=a/x- h+ k
We need to have the asymptotes of the functions to graph them. In this form, the asymptotes are the lines x= h and y= k. Let's now identify the values of h and k for our functions f(x) and y.
f(x)=7/x &⇔ f(x)=7/x- 0+ 0
y=7/x+3-2 &⇔ y=7/x-( -3)+( -2)
We can see that h= 0 and k= 0 for f(x). Therefore, the asymptotes are the lines x= 0 and y= 0. Moreover, h= -3 and k= -2 for y. This means that the asymptotes are the lines x= -3 and y= -2.
&Asymptotes
f(x): &⇒ x= 0 & y= 0
y: &⇒ x= -3 & y= -2
With this in mind, we will make a table of values to identify the points while f(x) is translated to y. To do so we will split the function y into two to determine the vertical and horizontal translations separately.
f(x)=7/x
7/x+3
y=7/x+3-2
x
7/x
(x,y)
x
7/x+3
(x,y)
x
7/x+3-2
(x,y)
- 3.5
7/- 3.5
( - 3.5, - 2)
- 6.5
7/-6.5+3
( -6.5, -2)
- 6.5
7/-6.5+3-2
( -6.5, -4)
- 2
7/- 2
( -2, - 3.5)
-5
7/-5+3
( -5, - 3.5)
-5
7/-5+3-2
( -5, - 5.5)
- 1
7/- 1
( - 1, - 7)
-4
7/-4+3
( -4, - 7)
-4
7/-4+3-2
( -4, - 9)
1
7/1
( 1, 7)
-2
7/-2+3
( -2, 7)
-2
7/-2+3-2
( -2, 5)
2
7/2
( 2, 3.5)
-1
7/-1+3
( -1, 3.5)
-1
7/-1+3-2
( -1, 1.5)
3.5
7/3.5
( 3.5, 2)
0.5
7/0.5+3
( 0.5, 2)
0.5
7/0.5+3-2
( 0.5, 0)
Let's now plot and connect the obtained points for each function on the same coordinate plane to identify the translations. Remember that both of the graphs will have two branches.
As we can see, the graph of the function y is a 3-unit translation of the graph of f(x) to the left and a 2-unit translation down.
