Find the End Behavior q(x)=3x^7+5x^5-8x^4-12x^2

$q (x) = 3 x^{7} + 5 x^{5} - 8 x^{4} - 12 x^{2}$

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.

$3 x^{7} \to 7$

$5 x^{5} \to 5$

$- 8 x^{4} \to 4$

$- 12 x^{2} \to 2$

The largest exponent is the degree of the polynomial.

$7$

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.

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

$3 x^{7}$

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

$3$

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.

Falls to the left and rises to the right

Step 7