Question

Describe the long run behavior of f(p)=(p+1)^3(p+4)^3(p-1)

Answer 1

Step-by-step explanation:

The long-run behavior of a function can be determined by examining its asymptotes and the behavior of the function near these asymptotes.

For the function f(p)=(p+1)^3(p+4)^3(p-1), there are three points to consider as possible asymptotes: p = -1, p = -4, and p = 1.

At p = -1, the factor (p-1) is equal to zero, so the function has a vertical asymptote there. This means that as p approaches -1 from either side, the function will approach infinity.

At p = -4, the factor (p+4)^3 is equal to zero, so the function also has a vertical asymptote there. Similarly, as p approaches -4 from either side, the function will approach infinity.

At p = 1, the factor (p+1)^3 is equal to zero, so the function has another vertical asymptote there. In this case, as p approaches 1 from either side, the function will approach negative infinity.

So, the long-run behavior of the function can be described as follows:

The function approaches infinity as p approaches -1 or -4 from either side.

The function approaches negative infinity as p approaches 1 from either side.

Therefore, the long-run behavior of f(p)=(p+1)^3(p+4)^3(p-1) is characterized by three vertical asymptotes at p = -1, p = -4, and p = 1.

Answer 2

`\text{As } p \to -\infty, f(p) \to -\infty\\\\\text{As } p \to \infty, f(p) \to \infty`

====================================================

Step-by-step explanation:

Each factor is a cubic of degree 3.

The degree of the entire polynomial is 3+3+3 = 9.

Since the degree is odd and the leading coefficient is positive, this means the graph falls to the left and rises to the right.

"falls to the left" means
`\text{As } p \to -\infty, f(p) \to -\infty`

"rises to the right" means
`\text{As } p \to \infty, f(p) \to \infty`

This is verified using a graphing tool like Desmos as shown in the screenshot below. I'm using x in place of p.