How can you create a function based on the given features: a vertical asymptote at y=2, a horizontal asymptote at x=3, a y-intercept at (0, -4/3) and an x-intercept at (-2,0)?

Answer 1

Y asymptote=leading coefficient of N(x)/leading coefficient of D(x)=2

X asymptote=3, thus when D(x)=0, then x=3, thus D(x)=x-3

f(0)=-4/3

f(-2)=0, thus when N(x)=0, then x=-2, thus N(x)=x+2

f(x)= (Y asymptote) (x+2)/(x-3)

f(x)=2(x+2)/(x-3)

You can see that the last condition would be satisfied too: when x=0, y would be -4/3

Answer 2

First create a general function. This is going to be a rational function, since we have vertical and horizontal asymptotes. Also the degree in the numerator, must be equal to the degree in the denominator. One possible general function could look like this:

y = a (x-p)/(x-q). We would then just solve for the three variables a, p, and q. However, this might not work because we only have 3 variables but 4 bits of information from the problem (2 asymptotes and 2 intercepts).

Thus, we need to include more variables.

The general function we'll use is:

y = a(x-p)(x-q)/[(x-r)(x-s)] The numerator and denominator have a degree of 2 and this is rational.

Now, solve for a, p, q, r, and s.

There is a horizontal asymptote at y= 2. Therefore, a = 2

There is an x-intercept at (-2,0). Therefore, either p or q = -2. We'll say that p = -2

There is a vertical asymptote at x = 3. Therefore, either r or s = 3. We'll say that r = 3.

Let's plug in what we have so far:

y = 2(x+2)(x-q)/[(x-3)(x-s)]

The last bit of information says that we have a y-intercept at (0,-4/3). So we'll plug in 0 for x and -4/3 for y.

When you do this and simplify, you'll get: q/s = 1.

Therefore, q = s. If q = s, then (x-q) on the top and (x-s) on the bottom will cancel out.

And the equation we'll have is y = 2(x+2)/(x-3)