Find a rational equation in factored form with:
- x-intercepts at x= 3 and x=-1
- y-intercept at y= 1/6
- horizontal asymptote at y=1
- Vertical asymptote at x=2 and x= -3
vertical asymptotes:
y = 1/ (x-2)(x-3)
x-intercepts:
y = (x-3)(x+1) / (x-2)(x-3)
that has a horizontal asymptote at y=1
Now, how about that pesky y-intercept at (0,1/6)?
How will you fix that up?
Oops. Excuse my typo. It should have said
y = (x-3)(x+1) / (x-2)(x+3)
Now, we need to move the y-intercept, without moving the horizontal asymptote. How can we multiply y by 1/3 when x=0, but not change y for large x? This will do the trick:
y = (x-3)(x+1) / (x-2)(x-3) * (x^2+1)/(x^2+3)
that does not move any of the other features required, but does change the y-intercept. See the graph at
https://www.wolframalpha.com/input/?i=%28%28x-3%29%28x%2B1%29%29+%2F+%28%28x-2%29%28x%2B3%29%29+*+%28x%5E2%2B1%29%2F%28x%5E2%2B3%29
To find a rational equation in factored form that satisfies these conditions, follow these steps:
Step 1: Start with the general form of a rational equation in factored form:
f(x) = A * (x - r1)^m1 * (x - r2)^m2 * ... * (x - rn)^mn / (x - s1)^n1 * (x - s2)^n2 * ... * (x - sn)^nn
In this form, A represents a constant, r1, r2, ..., rn represent the x-intercepts, s1, s2, ..., sn represent the vertical asymptotes, and m1, m2, ..., mn, n1, n2, ..., nn represent the multiplicities of the corresponding intercepts/asymptotes.
Step 2: Use the given x-intercepts to form the factors of the numerator:
(x - 3)(x + 1)
Step 3: Use the given y-intercept to determine the value of A:
To have a y-intercept (0, 1/6), substitute x = 0 and y = 1/6 into the equation:
1/6 = A * (0 - 3)(0 + 1)
1/6 = -3A
A = -1/18
Step 4: Use the given asymptotes to form the factors of the denominator:
(x - 2)(x + 3)
Step 5: Combine the factors in the numerator and denominator to form the final equation:
f(x) = (-1/18)(x - 3)(x + 1) / (x - 2)(x + 3)
Therefore, the rational equation in factored form with the given conditions is:
f(x) = (-1/18)(x - 3)(x + 1) / (x - 2)(x + 3)