Find the End Behavior f(x)=-216x^4+8x^6+216x^2+9720-7776x+1152x^3-32x^5

$f (x) = - 216 x^{4} + 8 x^{6} + 216 x^{2} + 9720 - 7776 x + 1152 x^{3} - 32 x^{5}$

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.

$- 216 x^{4} \to 4$

$8 x^{6} \to 6$

$216 x^{2} \to 2$

$9720 \to 0$

$- 7776 x \to 1$

$1152 x^{3} \to 3$

$- 32 x^{5} \to 5$

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 $9720$ .

$- 216 x^{4} + 8 x^{6} + 216 x^{2} - 7776 x + 1152 x^{3} - 32 x^{5} + 9720$

Move $- 7776 x$ .

$- 216 x^{4} + 8 x^{6} + 216 x^{2} + 1152 x^{3} - 32 x^{5} - 7776 x + 9720$

Move $216 x^{2}$ .

$- 216 x^{4} + 8 x^{6} + 1152 x^{3} - 32 x^{5} + 216 x^{2} - 7776 x + 9720$

Move $1152 x^{3}$ .

$- 216 x^{4} + 8 x^{6} - 32 x^{5} + 1152 x^{3} + 216 x^{2} - 7776 x + 9720$

Move $- 216 x^{4}$ .

$8 x^{6} - 32 x^{5} - 216 x^{4} + 1152 x^{3} + 216 x^{2} - 7776 x + 9720$

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

$8 x^{6}$

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

$8$

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)=-216x^4+8x^6+216x^2+9720-7776x+1152x^3-32x^5

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