Find the End Behavior f(x)=3456+114x^3-948x^2-6x^5+48x^4-144x

$f (x) = 3456 + 114 x^{3} - 948 x^{2} - 6 x^{5} + 48 x^{4} - 144 x$

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.

$3456 \to 0$

$114 x^{3} \to 3$

$- 948 x^{2} \to 2$

$- 6 x^{5} \to 5$

$48 x^{4} \to 4$

$- 144 x \to 1$

The largest exponent is the degree of the polynomial.

$5$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Move $3456$ .

$114 x^{3} - 948 x^{2} - 6 x^{5} + 48 x^{4} - 144 x + 3456$

Move $- 948 x^{2}$ .

$114 x^{3} - 6 x^{5} + 48 x^{4} - 948 x^{2} - 144 x + 3456$

Move $114 x^{3}$ .

$- 6 x^{5} + 48 x^{4} + 114 x^{3} - 948 x^{2} - 144 x + 3456$

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

$- 6 x^{5}$

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

$- 6$

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.

Rises to the left and falls to the right

Step 7