Write the equation of a Rational Function satisfying given conditions.
Has vertical asymptotes located at x-2 and x=-1
Has a horizontal asymptote located at y=0(x-axis)
y-intercept:(0,2) x-intercept (4,0)
R(x)= ???
what I did is,
R(x)= ax+b/c(x-2)(x+1)
then plug in (0,2), b=-4c
plug in(4,0), a=c
then what to do next?? Please help me with this I will appreciate that!!
You don't need any c. It can be absorbed into a and b.
Actually, you have solved the problem. Pick any value for c, and plug it in. As long as a=c and b=-4c, R(x) will work. So, make things easy. Let c=1.
As I worked it out,
R(x) = ???/(x+1)(x-2)
Since y=0 is the asymptote, you know that the degree of the numerator is less than the denominator. So,
R(x) = (ax+b) / (x+1)(x-2)
Now, with the intercepts, you know that
b/(1)(-2) = 2, so b = -4
a(4)+b = 0, so 4a-4=0, so a=1
R(x) = (x-4)/(x+1)(x-2)
To find the equation of a rational function, given the conditions you mentioned, we can follow these steps:
Step 1: Determine the form of the equation:
Since the function has two vertical asymptotes at x = 2 and x = -1, the denominator of the rational function will be (x - 2)(x + 1).
The numerator can be represented as ax + b, where a and b are constants.
Therefore, the equation takes the form R(x) = (ax + b) / ((x - 2)(x + 1)).
Step 2: Use the given conditions to solve for the values of a and b:
a. Using the y-intercept (0, 2):
When x = 0, R(x) = 2.
Substituting these values into the equation, we have:
(0a + b) / ((0 - 2)(0 + 1)) = 2
b/-2 = 2
b = -4
b. Using the x-intercept (4, 0):
When R(x) = 0, x = 4.
Substituting these values into the equation, we have:
(4a - 4)/((4 - 2)(4 + 1)) = 0
4a - 4 = 0
4a = 4
a = 1
Therefore, a = 1 and b = -4.
Step 3: Substitute the values of a and b back into the equation:
R(x) = (1x - 4) / ((x - 2)(x + 1))
Simplifying further:
R(x) = (x - 4) / ((x - 2)(x + 1))
So, the equation of the rational function satisfying the given conditions is:
R(x) = (x - 4) / ((x - 2)(x + 1))