Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Leading Coefficient Test to determine the right-hand and left-hand behavior of the graph of the polynomial function. $$f(x)=5-x^{2}-4 x^{4}$$

Step-by-Step Solution

Answer

Using the Leading Coefficient Test, it can be stated that the graph of the function \(f(x)=5-x^{2}-4 x^{4}\) falls or ends facing downwards on both the left-hand and right-hand side.

Step 1: Determine the leading coefficient and the polynomial's degree

The leading coefficient is the coefficient of the term with the highest power, and the degree is the highest power of the polynomial. In this function \(f(x)=5-x^{2}-4 x^{4}\), -4 is the leading coefficient and 4 is the degree.

Step 2: Apply the Leading Coefficient Test

The Leading Coefficient Test tells you about the ends of a polynomial's graph based on the degree and the sign of the leading coefficient. Given that we have a leading coefficient of -4 (which is negative) and a degree of 4 (which is even), a negative leading coefficient means 'down' and an even degree also means 'down' according to the Leading Coefficient Test. Therefore, as \(x \rightarrow \pm \infty\) (i.e., to the far right or left), the graph also 'goes down', or decreases.

Step 3: Conclusion

According to the Leading Coefficient Test, it can be concluded that the graph of the polynomial \(f(x)=5-x^{2}-4 x^{4}\) ends facing downwards on both the left-hand side (as \(x\rightarrow -\infty\)) and the right-hand side (as \(x\rightarrow +\infty\)). This is the end behavior of the polynomial.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial function is given by:
\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + \.\.\. + a_2x^2 + a_1x + a_0 \]
where \(a_n, a_{n-1}, ..., a_1, a_0\) are constants known as coefficients, and \(n\) is a non-negative integer. The power of \(x\), represented by \(n\), corresponds to the term's degree. Polynomial functions are classified based on the number of terms they have (monomial, binomial, trinomial) or their highest degree.
For example, the function \(f(x)=5-x^{2}-4x^{4}\) is a polynomial function where \(5\), \(-x^{2}\), and \(-4x^{4}\) are individual terms. It's important to note that in the world of algebra, these functions are fundamental, often used to model various physical phenomena and solve a wide range of problems due to their nice properties, such as continuity and differentiability.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in its expression. It holds significant information about the polynomial, such as the number of roots it can have and the behavior of the graph. When determining the degree, it's crucial to look for the term with the highest exponent.
In our example, \(f(x)=5-x^{2}-4x^{4}\), the term with the highest exponent is \(-4x^{4}\); hence, the degree is 4. Knowing the degree of a function allows for predictions concerning the rough shape of the graph and the number of turns it can take. Polynomials of even degrees tend to have similar end behaviors at both extremities, while those of odd degrees usually have opposite end behaviors.
Understanding the degree of a polynomial is essential in algebra and is a major step in graph analysis and the interpretation of the possible real-life applications of these mathematical expressions.

End Behavior of Polynomials

The end behavior of polynomials describes how the values of the polynomial function behave as the input values become very large or very small; specifically, as \(x\) approaches infinity or negative infinity. It provides a description of the tails of the graph.
The Leading Coefficient Test is a quick way to determine the end behavior. It uses the degree of the polynomial and the sign of the leading coefficient (the coefficient of the term with the highest power) to predict this behavior.

• For a polynomial with an even degree and a positive leading coefficient, the ends of the graph will face upwards.
• For a polynomial with an even degree and a negative leading coefficient, like our function \(f(x)=5-x^{2}-4x^{4}\), both ends face downwards.
• For a polynomial with an odd degree and a positive leading coefficient, the left end will face downwards and the right end will face upwards.
• For a polynomial with an odd degree and a negative leading coefficient, the left end will face upwards and the right end will face downwards.

Analyzing the end behavior is instrumental for sketching the rough graph of the function and understanding its long-range behavior, which in many cases can be crucial for predicting the outcome of real-world scenarios modeled by polynomials.