A rational function of the form y = a/(x - h) + k has asymptotes at x = 4 and y = - 2.
Write the equations of two different functions that meet this description.
I am completely blank on this! Please help.
in your equation y = a/(x - h) + k
a vertical asymptote is caused by the x-h, it can't be zero
so the h must be 4
a horizontal asymptote is caused when the term a/(x-4) gets closer and closer to zero
then you would be left with y = 0 + k
so k = -2
your equation could be y = a/(x-4) - 2
we know nothing about a, so it could be anything
e.g.
y = 3/(x-4) - 2
y = -5/(x-4) - 2
notice in the graphs shown, both have x=4 as a vertical asymptote and
y = -2 as a horizontal asymptote
btw, if a = 0 , we would get the horizontal line of y = -2 with a hole at (4,-2)
To find the equations of the two different rational functions, we need to use the information given about the asymptotes and then determine the values of the parameters a, h, and k.
1. Asymptote at x = 4:
For a rational function y = a/(x - h) + k, if there is an asymptote at x = 4, it means that the denominator (x - h) should be equal to zero at x = 4. So, we have:
x - h = 4
h = x - 4
2. Asymptote at y = -2:
For a rational function y = a/(x - h) + k, if there is an asymptote at y = -2, it means that when x approaches infinity, y approaches -2. This implies that k = -2.
Now, we have the following equations:
h = x - 4
k = -2
We can use these equations to create two different functions:
Function 1:
Let's choose a value for a, say a = 1. Putting this value into the rational function, we get:
y = 1/(x - (x - 4)) - 2
Simplifying this expression, we have:
y = 1/4 - 2
y = -7/4
Therefore, one equation that satisfies the given conditions is: y = -7/4.
Function 2:
Let's choose a different value for a, say a = -3. Putting this value into the rational function, we get:
y = -3/(x - (x - 4)) - 2
Simplifying this expression, we have:
y = -3/4 - 2
y = -11/4
Therefore, another equation that satisfies the given conditions is: y = -11/4.
So, the two different functions that meet the given description are:
Function 1: y = -7/4
Function 2: y = -11/4
To find the equations of two different rational functions that have asymptotes at x = 4 and y = -2, we need to determine the values of a, h, and k.
We know that the equation of a vertical asymptote is in the form x = h, and the equation of a horizontal asymptote is in the form y = k.
Since the vertical asymptote is x = 4, we have h = 4.
Since the horizontal asymptote is y = -2, we have k = -2.
Now, let's consider different values for a to create two different rational functions. Let's say a = 2 and a = -3.
First Function:
Using a = 2, h = 4, and k = -2, the rational function becomes:
y = 2/(x - 4) - 2
Second Function:
Using a = -3, h = 4, and k = -2, the rational function becomes:
y = -3/(x - 4) - 2
These are two different functions that satisfy the given conditions.