Give an example of a rational function that has vertical asymptote x = 3 and x = -3, horizontal asymptote y = 2 and y-intercept is (0, 4)
f(x) = (x-4)/(x^2-9) + 2
To find a rational function with the given characteristics, we can start by considering the vertical asymptotes. Since the vertical asymptotes are at x = 3 and x = -3, we know that the rational function will have factors of (x - 3) and (x + 3) in the denominator.
Let's assume the function is of the form:
f(x) = (x - 3)(x + 3)(ax + b) / (cx + d),
where a, b, c, and d are constants that we need to determine.
Next, let's consider the horizontal asymptote. The horizontal asymptote is y = 2, which means that as x approaches positive or negative infinity, the function approaches 2. To satisfy this condition, the degree of the numerator should be equal to or less than the degree of the denominator.
Since the highest degree of the denominator is 1, the highest degree of the numerator can also be 1. Therefore, let's assume the numerator is in the form of ax + b.
Now, let's consider the y-intercept. The y-intercept is (0, 4), which means when x = 0, f(x) = 4. Plugging these values into the rational function, we have:
f(0) = (0 - 3)(0 + 3)(a(0) + b) / (c(0) + d) = 4.
-9b / d = 4.
Taking this condition into account, we can let b = 4d/9.
Substituting the numerator and denominator forms into the rational function, we have:
f(x) = (x - 3)(x + 3)(ax + (4d/9)) / (cx + d).
Now, we have found expressions for both the numerator and denominator of the rational function with the given characteristics.
To find a rational function that satisfies the given conditions, we need to consider its equation in the form:
f(x) = (Numerator) / (Denominator)
Let's start by finding the denominator, which will determine the location of the vertical asymptotes. We know that the vertical asymptotes are at x = 3 and x = -3, so the denominator should have factors of (x - 3) and (x + 3). Therefore, the denominator could be:
Denominator: (x - 3)(x + 3)
Next, we need to find the numerator so that the function has the desired horizontal asymptote and y-intercept. The horizontal asymptote is given as y = 2, which means that the degree of the numerator and the denominator should be the same. To achieve this, let's consider a numerator of degree 1, which will also allow us to easily find the y-intercept.
Numerator: 2x + 4
Now, we have the numerator and the denominator of the rational function:
f(x) = (2x + 4) / ((x - 3)(x + 3))
This rational function satisfies the given conditions: it has vertical asymptotes at x = 3 and x = -3, a horizontal asymptote at y = 2, and its y-intercept is (0, 4).
Hmmm. f(0) = -4/-9 + 2 ≠ 4
How about
y = -18/(x^2-9) + 2 = 2(x^2-18)/(x^2-9)
That also has two x-intercepts. If you don't want any of those, then you will need to have y>0 for all x.
y = 2*9^2/(x^2-9)^2 + 2