Question

What is the end behavior of the graph of the polynomial function f(x) = 3x + 30x + 75x4?

Answer 1

Answer: As $x \to -\infty, y \to \infty$ and as $x \to \infty, y \to \infty$

Basically, whether x goes to positive or negative infinity, the value of y will head off to positive infinity.

Why is this? It's all because of the leading term 75x^4. This is the term with the largest exponent, so the degree is 4. Even degree functions have the endpoints pointing in the same direction. In this case, that direction is up or pointing along the positive y axis direction. If the leading coefficient was negative, then y would head off to negative infinity instead of positive infinity.