Find the End Behavior g(x)=5x^6+x^5+9x^3-12x-125

$g (x) = 5 x^{6} + x^{5} + 9 x^{3} - 12 x - 125$

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.

$5 x^{6} \to 6$

$x^{5} \to 5$

$9 x^{3} \to 3$

$- 12 x \to 1$

$- 125 \to 0$

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.

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

$5 x^{6}$

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

$5$

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