Find the End Behavior f(x)=800x^2-480x^3+2x^6-28x^5+160x^4+256-704x

$f (x) = 800 x^{2} - 480 x^{3} + 2 x^{6} - 28 x^{5} + 160 x^{4} + 256 - 704 x$

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.

$800 x^{2} \to 2$

$- 480 x^{3} \to 3$

$2 x^{6} \to 6$

$- 28 x^{5} \to 5$

$160 x^{4} \to 4$

$256 \to 0$

$- 704 x \to 1$

The largest exponent is the degree of the polynomial.

$6$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 3

Identify the leading coefficient.

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

Move $256$ .

$800 x^{2} - 480 x^{3} + 2 x^{6} - 28 x^{5} + 160 x^{4} - 704 x + 256$

Move $800 x^{2}$ .

$- 480 x^{3} + 2 x^{6} - 28 x^{5} + 160 x^{4} + 800 x^{2} - 704 x + 256$

Move $- 480 x^{3}$ .

$2 x^{6} - 28 x^{5} + 160 x^{4} - 480 x^{3} + 800 x^{2} - 704 x + 256$

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

$2 x^{6}$

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

$2$

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.

Rises to the left and rises to the right

Step 7