Question

How do you describe the end behavior for `f(x)=-x^5+4x^3-2x-2`?

Answer 1

Left side up, right side down

Explanation:

There are two steps to describing the end behavior of a polynomial:

• Determine the behavior of the right (`+x` "side") of the graph
• Determine the behavior of the left (`-x` "side") of the graph

Right Side Behavior

Determining the right side behavior of the graph involves looking at the leading coefficient, or the coefficient of the highest power of `x` term in the polynominal. In this case, the highest power `x` term is `-x^5`. Notably, this term has a negative leading coefficient (-1).

If the leading coefficient is negative, then the right edge of the graph points downward toward the negative `y` portion of the coordinate plane. Conversely, you can say that the function itself is "falling" or "decreasing" in value as you head off the right edge of the graph. More precisely, we'd say that as `x->+oo`, `f(x)->-oo`

For this problem, we say the right edge is pointing downward.

Left Side Behavior

Determining the left side behavior of the graph involves looking at the highest power of `x` in the polynomial. If that power/exponent is even, then the left side behaves the same as the right side; if that power/exponent is odd, then the left side behaves the opposite to the right. (I like to think the phrase even is equal, odd is opposite in my head.)

For this problem, the highest exponent is 5 (from `-x^5`), which is an odd power, which indicates the left side is opposite to the right. Thus, the left edge of the graph is pointing upward.

Check with the graph

You can verify with the graph:

graph{-x^5+4x^3-2x-2 [-3.24, 3.23, -7.12, 7.12]}