Let's start by reviewing the , y=x1. This function has two branches and its lie on the x- and y-axes.
Its is all except for
x=0 where the function is undefined. Similarly, the is all real values except for
0, since there is no
x-value for which
x1=0. Is for these reason that the graph has asymptotic behavior. We can our function vertically by
adding the constant
k to the parent function.
y=x1→y2=x1+k
If
k is positive, the function gets translated upwards by
k units. But if
k<0 then it gets translated downwards by
∣k∣ units. Note that, just as the function, the horizontal asymptote translates either down or up, accordingly. We can find an example below.
With this translation, the asymptotic behavior of the function around the horizontal asymptote gets shifted, and now the range is all real numbers except
3 — since there is no
x-value for which
y2=3.
FunctionRangey2=x1+kAll real numbers but3
Similarly, we can translate the function horizontally by
replacing the
x variable with
x−h in the parent function.
y=x1→y3=x−h1
If
h is positive, the function gets translated
h units to the right. But, if
h<0 it gets translated
∣h∣ units to the left. Just as in the previous example, if the function is translated horizontally, to the right of to the left, its vertical asymptote is translated in the same way. We can see an example below.
Because the vertical asymptote is shifted
3 units to the right, the new domain is all real numbers except
3, where the function is now undefined.
FunctionDomainy3=x−31All real numbers but3
We can also perform both translations at once. In this case, both the domain and range are modified in the same way we already discussed above.
y4=x−h1+kDomainRangeAll real numbers butAll real numbers buthk
In this exercise we are asked to describe how the domain and range change when
y=x1 gets translated
3 units up and
5 units to the left. To do this, we can just substitute
k=3 and
h=-5. Following the ideas above and using the values mentioned we can find the domain and range for the translated function.
y4=x−(-5)1+3⇔y4=x+51+3DomainRangeAll real numbersAll real numbersexcept -5except 3