Find the End Behavior f(x)=-738x^3+18x^5+4392x+159x^4-1884x^2-3x^6+8640

$f (x) = - 738 x^{3} + 18 x^{5} + 4392 x + 159 x^{4} - 1884 x^{2} - 3 x^{6} + 8640$

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.

$- 738 x^{3} \to 3$

$18 x^{5} \to 5$

$4392 x \to 1$

$159 x^{4} \to 4$

$- 1884 x^{2} \to 2$

$- 3 x^{6} \to 6$

$8640 \to 0$

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 $4392 x$ .

$- 738 x^{3} + 18 x^{5} + 159 x^{4} - 1884 x^{2} - 3 x^{6} + 4392 x + 8640$

Move $- 1884 x^{2}$ .

$- 738 x^{3} + 18 x^{5} + 159 x^{4} - 3 x^{6} - 1884 x^{2} + 4392 x + 8640$

Move $- 738 x^{3}$ .

$18 x^{5} + 159 x^{4} - 3 x^{6} - 738 x^{3} - 1884 x^{2} + 4392 x + 8640$

Move $159 x^{4}$ .

$18 x^{5} - 3 x^{6} + 159 x^{4} - 738 x^{3} - 1884 x^{2} + 4392 x + 8640$

Reorder $18 x^{5}$ and $- 3 x^{6}$ .

$- 3 x^{6} + 18 x^{5} + 159 x^{4} - 738 x^{3} - 1884 x^{2} + 4392 x + 8640$

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

$- 3 x^{6}$

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

$- 3$

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

Find the End Behavior f(x)=-738x^3+18x^5+4392x+159x^4-1884x^2-3x^6+8640

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