Find the End Behavior 3x^6+30x^5+75x^4

$3 x^{6} + 30 x^{5} + 75 x^{4}$

Step 1

Write $3 x^{6} + 30 x^{5} + 75 x^{4}$ as a function.

$f (x) = 3 x^{6} + 30 x^{5} + 75 x^{4}$

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.

$3 x^{6} \to 6$

$30 x^{5} \to 5$

$75 x^{4} \to 4$

The largest exponent is the degree of the polynomial.

$6$

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.

$3 x^{6}$

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.

Rises to the left and rises to the right

Step 8