Find the End Behavior f(x)=-10x^6+5120-400x^3-2240x^2+1280x+20x^5+280x^4

$f (x) = - 10 x^{6} + 5120 - 400 x^{3} - 2240 x^{2} + 1280 x + 20 x^{5} + 280 x^{4}$

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.

$- 10 x^{6} \to 6$

$5120 \to 0$

$- 400 x^{3} \to 3$

$- 2240 x^{2} \to 2$

$1280 x \to 1$

$20 x^{5} \to 5$

$280 x^{4} \to 4$

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 $5120$ .

$- 10 x^{6} - 400 x^{3} - 2240 x^{2} + 1280 x + 20 x^{5} + 280 x^{4} + 5120$

Move $1280 x$ .

$- 10 x^{6} - 400 x^{3} - 2240 x^{2} + 20 x^{5} + 280 x^{4} + 1280 x + 5120$

Move $- 2240 x^{2}$ .

$- 10 x^{6} - 400 x^{3} + 20 x^{5} + 280 x^{4} - 2240 x^{2} + 1280 x + 5120$

Move $- 400 x^{3}$ .

$- 10 x^{6} + 20 x^{5} + 280 x^{4} - 400 x^{3} - 2240 x^{2} + 1280 x + 5120$

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

$- 10 x^{6}$

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

$- 10$

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