Find an equation of a rational function with the following characteristics:
x-int of 5, y-int of -5/8, vertical asymptote x=-8/5, horizontal asymptote y=1/3
What is a possible answer and how did you arrive at each step?
vertical asymptote
y = a/(5x+8)
x-int
y = a(x-5)/(5x+8)
horizontal asymptote
y = (5/3)(x-5)/((5x+8)) = (5x-25)/(15x+24)
This has a y-intercept of -25/24
So, how will you adjust it to get -5/8?
you want y(0) = -5/8, so
How would I continue from there?
To find an equation of a rational function with the given characteristics, we can follow these steps:
Step 1: Determine the numerator degree and denominator degree
To find the equation of a rational function, we need to determine the degrees of the numerator and denominator. Since we know that the vertical asymptote is x = -8/5, the denominator will contain a factor of (x - (-8/5)) which simplifies to (x + 8/5) on factoring. This factor will be multiplied by some other factors. The numerator will contain a factor of (x - x-int), which is (x - 5) in this case.
Step 2: Determine the horizontal asymptote
The horizontal asymptote is y = 1/3. Since the degree of the numerator is less than or equal to the degree of the denominator, the horizontal asymptote will be given by the ratio of the leading coefficients of the numerator and denominator. In this case, the ratio is 1/3.
Step 3: Determine the y-intercept
The y-intercept is -5/8. This means that when x = 0, y is equal to -5/8. So, we can let the function have a constant term of -5/8.
Step 4: Write the equation of the rational function
Putting all this information together, the equation for the rational function is:
f(x) = k * (x - x-int) / [(x + 8/5) * (x - x1) * (x - x2) * ...]
where k is a constant, x-int is the x-intercept (5 in this case), and x1, x2, ... are any additional x-intercepts that may exist.
For this particular problem, the equation could be:
f(x) = (1/3) * (x - 5) / [(x + 8/5) * (x - 5)]
Please note that this is just one possible equation, as there could be other variations based on the given characteristics.