Find the End Behavior y=12x^4-x+3x^7-1

$y = 12 x^{4} - x + 3 x^{7} - 1$

Step 1

Simplify $12 x^{4} - x + 3 x^{7} - 1$ .

Move $- x$ .

$y = 12 x^{4} + 3 x^{7} - x - 1$

Reorder $12 x^{4}$ and $3 x^{7}$ .

$y = 3 x^{7} + 12 x^{4} - x - 1$

Step 2

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.

$3 x^{7} \to 7$

$12 x^{4} \to 4$

$- x \to 1$

$- 1 \to 0$

The largest exponent is the degree of the polynomial.

$7$

Step 3

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

Odd

Step 4

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 5

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 6

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 7

Determine the behavior.

Falls to the left and rises to the right

Step 8