Calculus Examples

$f (x) = x^{4} - 6$

Identify the degree of the function.

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Identify the term with the largest exponent on the variable.

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The degree is the sum of the exponents of each variable in the expression. In this case, the degree of $x^{4}$ is $4$ .

$4$

The degree is the sum of the exponents of each variable in the expression. In this case, the degree of $- 6$ is $0$ .

$0$

Identify the term with the largest exponent on the variable.

$x^{4}$

The degree of the polynomial is the largest exponent on the variable.

$4$

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

Even

Identify the leading coefficient.

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A polynomial consists of terms, which are also known as monomials. The leading term in a polynomial is the highest degree term. In this case, the leading term in $x^{4} - 6$ is the first term, which is $x^{4}$ .

$x^{4}$

The leading coefficient in a polynomial is the coefficient of the leading term. In this case, the leading term is $x^{4}$ and the leading coefficient is $1$ .

$1$

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

Positive

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

Determine the behavior.

Rises to the left and rises to the right

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