Find the Behavior (Leading Coefficient Test) f(x)=4x^4+4x^3+4x^2+5x+3

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

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.

$4 x^{4} \to 4$

$4 x^{3} \to 3$

$4 x^{2} \to 2$

$5 x \to 1$

$3 \to 0$

The largest exponent is the degree of the polynomial.

$4$

Step 2

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

Even

Step 3

Identify the leading coefficient.

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

$4 x^{4}$

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

$4$

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.

Rises to the left and rises to the right

Step 7