Find the End Behavior f(x)=-45x^5-2736x+288x^2+864x^3-9x^6-2880+144x^4

$f (x) = - 45 x^{5} - 2736 x + 288 x^{2} + 864 x^{3} - 9 x^{6} - 2880 + 144 x^{4}$

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.

$- 45 x^{5} \to 5$

$- 2736 x \to 1$

$288 x^{2} \to 2$

$864 x^{3} \to 3$

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

$- 2880 \to 0$

$144 x^{4} \to 4$

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 $- 2880$ .

$- 45 x^{5} - 2736 x + 288 x^{2} + 864 x^{3} - 9 x^{6} + 144 x^{4} - 2880$

Move $- 2736 x$ .

$- 45 x^{5} + 288 x^{2} + 864 x^{3} - 9 x^{6} + 144 x^{4} - 2736 x - 2880$

Move $288 x^{2}$ .

$- 45 x^{5} + 864 x^{3} - 9 x^{6} + 144 x^{4} + 288 x^{2} - 2736 x - 2880$

Move $864 x^{3}$ .

$- 45 x^{5} - 9 x^{6} + 144 x^{4} + 864 x^{3} + 288 x^{2} - 2736 x - 2880$

Reorder $- 45 x^{5}$ and $- 9 x^{6}$ .

$- 9 x^{6} - 45 x^{5} + 144 x^{4} + 864 x^{3} + 288 x^{2} - 2736 x - 2880$

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)=-45x^5-2736x+288x^2+864x^3-9x^6-2880+144x^4

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