Find the End Behavior x^3+2x^2+4x+5

$x^{3} + 2 x^{2} + 4 x + 5$

Step 1

Write $x^{3} + 2 x^{2} + 4 x + 5$ as a function.

$f (x) = x^{3} + 2 x^{2} + 4 x + 5$

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.

$x^{3} \to 3$

$2 x^{2} \to 2$

$4 x \to 1$

$5 \to 0$

The largest exponent is the degree of the polynomial.

$3$

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.

$x^{3}$

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.

Falls to the left and rises to the right

Step 8