Find the End Behavior y=10-3x^3+3x^2+x^4

$y = 10 - 3 x^{3} + 3 x^{2} + x^{4}$

Step 1

Simplify $10 - 3 x^{3} + 3 x^{2} + x^{4}$ .

Move $10$ .

$y = - 3 x^{3} + 3 x^{2} + x^{4} + 10$

Move $3 x^{2}$ .

$y = - 3 x^{3} + x^{4} + 3 x^{2} + 10$

Reorder $- 3 x^{3}$ and $x^{4}$ .

$y = x^{4} - 3 x^{3} + 3 x^{2} + 10$

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.

$x^{4} \to 4$

$- 3 x^{3} \to 3$

$3 x^{2} \to 2$

$10 \to 0$

The largest exponent is the degree of the polynomial.

$4$

Step 3

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 4

Identify the leading coefficient.

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

$x^{4}$

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

$1$

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.

Rises to the left and rises to the right

Step 8