how do you create a function that has a graph with the given features:
a. A vertical asymptote at x=2; a horizontal asymptote at y=0; no x-intercept; y-intercept is 3
To create a function with the given features, you can use rational functions.
A rational function is a function that can be expressed as the quotient of two polynomial functions. The general form of a rational function is f(x) = (p(x))/(q(x)), where p(x) and q(x) are polynomial functions.
Based on the given features, we can start by setting up the denominator (q(x)) of our rational function to have a vertical asymptote at x=2. This means that q(x) must have a factor of (x-2) in the denominator.
Now, to have a horizontal asymptote at y=0, the degree of the numerator (p(x)) must be less than or equal to the degree of the denominator (q(x)). In other words, the numerator should have a lesser or equal number of terms compared to the denominator.
To ensure there are no x-intercepts, we can set the numerator to be a constant function, meaning it only has one term that does not involve x. Thus, p(x) can simply be a constant c.
Finally, to satisfy the y-intercept at y=3, we need to set f(0) = 3. Substituting x=0 into the rational function, we get (c)/(q(0)) = 3. Therefore, q(0) must be equal to c/3.
Putting all of this together, a function that satisfies the given features is:
f(x) = c/((x-2)*3)
You can choose any non-zero constant value for c to get different functions that satisfy the given features.