Plot both of the graphs and then compare some points on graphs.
Translates the graph 12 units left
Practice makes perfect
We are given a rational function f(x)= 7x. We will find the needed translation to get the other function f(x)= 7x+12. 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. Since both of the functions have been named f, we will rename them as f_1(x) and f_2(x).
f_1(x)=7/x &⇔ f_1(x)=7/x- 0+ 0
f_2(x)=7/x+12 &⇔ f_2(x)=7/x-( -12)+ 0
We can see that h= 0 and k= 0 for f_1(x). Therefore, the asymptotes are the lines x= 0 and y= 0. Moreover, h= -12 and k= 0 for f_2(x). This means that the asymptotes are the lines x= -12 and y= 0. With this in mind, we will make a table of values for each function. Be sure to include numbers on both sides of the vertical asymptotes.
f_1(x)=7/x
f_2(x)=7/x+12
x
7/x
(x,y)
x
7/x+12
(x,y)
- 2
7/- 2
( -2, -3.5)
- 14
7/- 14+12
( - 14, -3.5)
- 1
7/- 1
( -1, -7)
- 13
7/- 13+12
( - 13, - 7)
-0.5
7/- 0.5
( -0.5, - 14)
- 12.5
7/- 12.5+12
( - 12.5, - 14)
0.5
7/0.5
( 0.5, 14)
- 11.5
7/- 11.5+12
( - 11.5, 14)
1
7/1
( 1, 7)
-11
7/-11+12
( -11, 7)
2
7/2
( 2, 3.5)
-10
7/-10+12
( -10, 3.5)
Let's now plot and connect the obtained points for each function on the same coordinate plane to identify the translation. Remember that both of the graphs will have two branches.
As we can see, the graph of f_2(x) is a 12-unit translation of the graph of f_1(x) to the left.
