Problem 27 - Precalculus : Building Concepts and Connections

Question

Determine the end behavior of the function. $$H(x)=-5 x^{4}+3 x^{2}+x-1$$

Step-by-Step Solution

Answer

The end behavior of the function \(H(x)=-5x^{4}+3x^{2}+x-1\) can be determined as both ends of the graph moving downwards. As \(x\) approaches positive or negative infinity, \(H(x)\) approaches negative infinity.

Step 1: Identify the leading term

The leading term of the polynomial function is the term with the highest degree. In \(H(x)=-5 x^{4}+3 x^{2}+x-1\), the leading term is \(-5x^{4}.\)

Step 2: Understand the significance of the leading term

In a polynomial function, the leading term is the term that has the most impact on the graph of the function as \(x\) increases indefinitely in either the positive or negative direction. This leading term can determine the end behavior of the polynomial.

Step 3: Determine the end behavior

The polynomial function \(H(x)=-5x^{4}+3x^{2}+x-1\) is a degree 4 polynomial with a negative leading coefficient. For even degree polynomials, both ends of the graph move in the same direction. If the leading coefficient is positive, both ends of the graph would move upwards and if the leading coefficient is negative, both ends would move downwards. So, the end behavior of \(H(x)\) is as \(x\) approaches positive or negative infinity, \(H(x)\) is going to negative infinity.

Key Concepts

Polynomial Functions

A polynomial function is one of the fundamental concepts in algebra and calculus. It is an expression consisting of variables, coefficients, and exponents added or subtracted together. Think of it as a mathematical sentence made up of terms that follow a particular pattern.

• Each term in a polynomial function is made up of a coefficient (a constant number) multiplied by a variable raised to a non-negative integer exponent.
• Polynomials are classified based on their degree, which is the highest power of the variable in the expression.
• Examples of polynomial functions include linear functions ( ax + b ), quadratic functions ( ax^2 + bx + c ), and cubic functions ( ax^3 + bx^2 + cx + d ).

Polynomial functions can model a wide range of natural phenomena and are used in various fields to describe relationships and predict outcomes. These functions can have different characteristics, such as turning points and end behavior, which are influenced by the coefficients and degrees of the terms.

Leading Term

The leading term of a polynomial function is crucial when discussing its end behavior and overall shape. It is the term with the highest degree exponent, and it plays a dominant role as the variable grows larger.

• The leading term gives us the quickest insight into the function's end behavior, as the influence of the lower-degree terms diminishes as the variable violates or surpasses certain limits.
• If the function is written in standard form, the leading term will be the first term you encounter when reading the expression from left to right.
• For instance, in the function \(H(x)=-5x^{4}+3x^{2}+x-1\), the leading term is \(-5x^{4}\), which indicates that as \(x\) becomes very large or very small, this term will have the most significant impact on \(H(x)\).

Understanding the leading term helps in predicting how the polynomial behaves at the extremes, helping in graphing and comprehending the function efficiently.

Degree of Polynomial

The degree of a polynomial is a vital characteristic that tells us much about the function's form and behavior. It is the largest exponent in its expression, indicating how many times the variable is used as a factor.

• The degree determines the maximum number of solutions (or roots) the polynomial can have and the potential number of turning points in its graph.
• For example, a quadratic function, which has a degree of 2, can have up to two solutions and typically has one peak or trough.
• In the equation \(H(x)=-5x^{4}+3x^{2}+x-1\), the degree is 4, indicating it can have up to four roots and three turning points.

The degree also influences the end behavior of the polynomial, especially when combined with the leading term's coefficient. For even-degree polynomials like our example, the direction of the end behavior largely depends on whether the leading coefficient is positive or negative. This establishes if the polynomial graph will rise or fall as \(x\) approaches infinity.