Find the End Behavior f(x)=-140x^4-3728x+3360-20x^5+580x^3+4x^6+520x^2

$f (x) = - 140 x^{4} - 3728 x + 3360 - 20 x^{5} + 580 x^{3} + 4 x^{6} + 520 x^{2}$

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.

$- 140 x^{4} \to 4$

$- 3728 x \to 1$

$3360 \to 0$

$- 20 x^{5} \to 5$

$580 x^{3} \to 3$

$4 x^{6} \to 6$

$520 x^{2} \to 2$

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 $3360$ .

$- 140 x^{4} - 3728 x - 20 x^{5} + 580 x^{3} + 4 x^{6} + 520 x^{2} + 3360$

Move $- 3728 x$ .

$- 140 x^{4} - 20 x^{5} + 580 x^{3} + 4 x^{6} + 520 x^{2} - 3728 x + 3360$

Move $580 x^{3}$ .

$- 140 x^{4} - 20 x^{5} + 4 x^{6} + 580 x^{3} + 520 x^{2} - 3728 x + 3360$

Move $- 140 x^{4}$ .

$- 20 x^{5} + 4 x^{6} - 140 x^{4} + 580 x^{3} + 520 x^{2} - 3728 x + 3360$

Reorder $- 20 x^{5}$ and $4 x^{6}$ .

$4 x^{6} - 20 x^{5} - 140 x^{4} + 580 x^{3} + 520 x^{2} - 3728 x + 3360$

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 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

Find the End Behavior f(x)=-140x^4-3728x+3360-20x^5+580x^3+4x^6+520x^2

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