Problem 28 - Precalculus : Building Concepts and Connections

Question

Determine the end behavior of the function. $$h(x)=5 x^{6}-3 x^{3}$$

Step-by-Step Solution

Answer

The function \(h(x)=5x^{6}-3x^{3}\) tends towards positive infinity as \(x\) approaches both positive and negative infinity.

Step 1: Identify the Leading Term of the Polynomial

The leading term of the polynomial function given \(\(h(x)=5x^{6}-3x^{3}\) is \(5x^{6}\). It is the term with the highest power of \(x\).

Step 2: Determine the End Behavior

Because the leading term is \(5x^{6}\), the end behavior of the function depends on this term. The power \(6\) is an even number. Thus, the function's end behavior is: as \(x\) approaches positive infinity, \(h(x)\) also approaches positive infinity (written \(x \to \infty, h(x) \to \infty\)). Similarly, as \(x\) approaches negative infinity, \(h(x)\) also approaches positive infinity (written \(x \to -\infty, h(x) \to \infty\)).

Key Concepts

Leading Term

The leading term of a polynomial is one of the key components to understand its behavior. It is the term with the highest degree or power of the variable involved. In the polynomial function given, \(h(x)=5x^6-3x^3\), the leading term is \(5x^6\) because it has the highest exponent, which is 6.

To identify a leading term, look for the term with the largest exponent in the polynomial function. This term will have the greatest impact on the graph of the polynomial as the values of \(x\) become very large or very small. It determines the end behavior of the polynomial, which refers to what happens to the value of the function as \(x\) goes towards positive or negative infinity.

• The coefficient in front of the leading term, in this case 5, affects the direction but not the general shape of the graph.
• Even if there are other terms, the leading term dominates the behavior at the extremes of \(x\).
• A positive coefficient with an even power means the ends of the graph will rise upwards.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums and products of variables raised to non-negative integer powers. They are a fundamental topic in algebra and have widespread applications in mathematical analysis and beyond. The basic structure of a polynomial function is the sum of terms, each consisting of a coefficient, a variable, and a non-negative integer exponent.

Key features of polynomial functions include:

• The degree of the polynomial, which is determined by the highest exponent in the expression.
• The leading coefficient, which is the coefficient of the leading term.
• The end behavior, largely influenced by the degree and the leading coefficient.

These functions exhibit certain behaviors based on the degree and leading coefficient:

• If the degree is even, both ends of the graph will point in the same direction.
• If the degree is odd, the ends will point in opposite directions.
• A positive leading coefficient causes the right end to point upward, while a negative leading coefficient causes it to point downward.

Power of a Polynomial

The power of a polynomial, also known as its degree, is another critical aspect that must be understood. It directly impacts the shape and behavior of the polynomial's graph.

In our problem, \(h(x)=5x^6-3x^3\), the highest power is 6, making this a 6th-degree polynomial.
The degree of a polynomial gives us valuable information:

• An even power, like 6, means the graph's end behaviors are symmetrical.
• A higher degree indicates that the graph can have more turning points and intersections with the \(x\)-axis, though the maximum number of these is often less than or equal to the degree of the polynomial.
• Understanding the degree helps in predicting the end behavior and potential shape of the graph.

The power or degree of the polynomial is essential in many calculations and has practical implications in fields such as physics, engineering, and economics, where such functions are used to model complex behaviors.