Find the End Behavior f(x)=1456x^2-4x^6+1160x^3+52x^4-40x^5-6400-4000x

$f (x) = 1456 x^{2} - 4 x^{6} + 1160 x^{3} + 52 x^{4} - 40 x^{5} - 6400 - 4000 x$

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.

$1456 x^{2} \to 2$

$- 4 x^{6} \to 6$

$1160 x^{3} \to 3$

$52 x^{4} \to 4$

$- 40 x^{5} \to 5$

$- 6400 \to 0$

$- 4000 x \to 1$

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 $- 6400$ .

$1456 x^{2} - 4 x^{6} + 1160 x^{3} + 52 x^{4} - 40 x^{5} - 4000 x - 6400$

Move $1456 x^{2}$ .

$- 4 x^{6} + 1160 x^{3} + 52 x^{4} - 40 x^{5} + 1456 x^{2} - 4000 x - 6400$

Move $1160 x^{3}$ .

$- 4 x^{6} + 52 x^{4} - 40 x^{5} + 1160 x^{3} + 1456 x^{2} - 4000 x - 6400$

Move $52 x^{4}$ .

$- 4 x^{6} - 40 x^{5} + 52 x^{4} + 1160 x^{3} + 1456 x^{2} - 4000 x - 6400$

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

$- 4 x^{6}$

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

$- 4$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

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 falls to the right

Step 7