describe functions end behavior f(x)= 10-3x^5-8x^3

Question

answer choices are -♾, ♾, 0 or an asymptote

(x2)

Answer 1

Given: f(x) = 10 - 3x^5 - 8x^3

Describe the end behavior.

Solution:

This is a 5th degree equation.

The coefficient of x^5 is negative.

If you put it in standard form,

it will have a negative leading coefficient.

f(x) = -3x^5 - 8x^3 + 10

Therefore, the end behavior

is like any odd-degree polynomial.

The simplest odd-degree polynomial

is a straight line.

The end behavior of this polynomial

is like the end behavior of a straight line

with a negative leading coefficient (slope).

Therefore, it's like a straight line, slanting down.

Since it's a higher-degree polynomial,

it just wiggles around more.

But, the far-end behavior

will be the same as a straight line

with a negative slope.

END BEHAVIOR:

As x approaches negative infinity,

the function approaches positive infinity.

As x approaches positive infinity,

the function approaches negative infinity.

Answer 2

For polynomial functions the end behavior is determined by the leading term. (This is because the highest power will dominate as you input larger and larger values.)

In this case your leading term is -3x^5, remember the leading term is the term that has the highest power.

As x approaches ∞, -3x^5 approaches -∞

As x approaches -∞, -3x^5 approaches ∞

So the end behavior for f(x) is:

As x approaches ∞, f(x) approaches -∞

As x approaches -∞, f(x) approaches ∞