Precalculus Examples

$f (x) = 10 x^{5} + 7 x^{4} + 5$

Step 1

Identify the degree of the function.

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Step 1.1

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$10 x^{5} \to 5$

$7 x^{4} \to 4$

$5 \to 0$

Step 1.2

The largest exponent is the degree of the polynomial.

$5$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

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Step 3.1

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

$10 x^{5}$

Step 3.2

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

$10$

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.

Falls to the left and rises to the right

Step 7