Find the End Behavior f(x)=8064+54x^4-648x^2-9x^6+6912x-1332x^3+81x^5

$f (x) = 8064 + 54 x^{4} - 648 x^{2} - 9 x^{6} + 6912 x - 1332 x^{3} + 81 x^{5}$

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.

$8064 \to 0$

$54 x^{4} \to 4$

$- 648 x^{2} \to 2$

$- 9 x^{6} \to 6$

$6912 x \to 1$

$- 1332 x^{3} \to 3$

$81 x^{5} \to 5$

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 $8064$ .

$54 x^{4} - 648 x^{2} - 9 x^{6} + 6912 x - 1332 x^{3} + 81 x^{5} + 8064$

Move $6912 x$ .

$54 x^{4} - 648 x^{2} - 9 x^{6} - 1332 x^{3} + 81 x^{5} + 6912 x + 8064$

Move $- 648 x^{2}$ .

$54 x^{4} - 9 x^{6} - 1332 x^{3} + 81 x^{5} - 648 x^{2} + 6912 x + 8064$

Move $- 1332 x^{3}$ .

$54 x^{4} - 9 x^{6} + 81 x^{5} - 1332 x^{3} - 648 x^{2} + 6912 x + 8064$

Move $54 x^{4}$ .

$- 9 x^{6} + 81 x^{5} + 54 x^{4} - 1332 x^{3} - 648 x^{2} + 6912 x + 8064$

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

$- 9 x^{6}$

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

$- 9$

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)=8064+54x^4-648x^2-9x^6+6912x-1332x^3+81x^5

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