Question

Create a function whose graph has the given characteristics. (There are many correct answers.) Vertical asymptote: \(x=-3\) Horizontal asymptote: None

Short answer

The function \(F(x) = x^2/(x+3)\) has the desired characteristics.

Step-by-step solution

Determine the Vertical Asymptote

A vertical asymptote occurs in a rational function where the denominator equals zero and the numerator does not. So, the denominator should be a factor that equals zero at \(x = -3\). A function that has this property is: \(x+3\). So, the denominator of the function is \(x+3\). F(x) can be written as: \(F(x) = N(x)/(x+3)\). Where \(N(x)\) is the numerator of the function.

Determine the Horizontal Asymptote

To not have a horizontal asymptote, the degree of the numerator of the function should be greater than the degree of the denominator. This way, as \(x \rightarrow \pm \infty\), the function will not approach any y-value. Let's choose the numerator to be a quadratic function, making its degree 2, which is higher than the degree of the denominator. Let's choose \(N(x) = x^2\). So, the function \(F(x) = x^2/(x+3)\) exhibits the desired characteristics.