Find the End Behavior f(x)=3x^5+672x^2+816x+384+264x^3+48x^4

$f (x) = 3 x^{5} + 672 x^{2} + 816 x + 384 + 264 x^{3} + 48 x^{4}$

Step 1

Identify the degree of the function.

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$3 x^{5} \to 5$

$672 x^{2} \to 2$

$816 x \to 1$

$384 \to 0$

$264 x^{3} \to 3$

$48 x^{4} \to 4$

The largest exponent is the degree of the polynomial.

$5$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Move $384$ .

$3 x^{5} + 672 x^{2} + 816 x + 264 x^{3} + 48 x^{4} + 384$

Move $816 x$ .

$3 x^{5} + 672 x^{2} + 264 x^{3} + 48 x^{4} + 816 x + 384$

Move $672 x^{2}$ .

$3 x^{5} + 264 x^{3} + 48 x^{4} + 672 x^{2} + 816 x + 384$

Move $264 x^{3}$ .

$3 x^{5} + 48 x^{4} + 264 x^{3} + 672 x^{2} + 816 x + 384$

The leading term in a polynomial is the term with the highest degree.

$3 x^{5}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$3$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Falls to the left and rises to the right

Step 7