Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=x^{4}-5 x^{2} $$

Step-by-Step Solution

Answer

The end behavior: As \( x \to -\infty \), \( f(x) \to +\infty \) and as \( x \to +\infty \), \( f(x) \to +\infty \).

Step 1: Identify the Leading Term

The leading term in a polynomial function determines the end behavior of the function. For the function \( f(x) = x^4 - 5x^2 \), the leading term is \( x^4 \). This is because it has the highest power of \( x \).

Step 2: Determine Degree and Leading Coefficient

The degree of \( f(x) \) is 4, which is even. The leading coefficient (coefficient of the leading term \( x^4 \)) is 1, which is positive. These two facts will help us predict the end behavior of the function.

Step 3: Analyze End Behavior Using Polynomial Rules

For polynomials, if the degree is even and the leading coefficient is positive, as \( x \to -\infty \), \( f(x) \to +\infty \) and as \( x \to +\infty \), \( f(x) \to +\infty \). This means both ends of the graph will rise.

Step 4: Confirm with a Table of Values

Make a table with several x-values, including negative and positive extremes:\[\begin{array}{|c|c|}\hlinex & f(x) = x^4 - 5x^2 \\hline-3 & (-3)^4 - 5(-3)^2 = 81 - 45 = 36 \-1 & (-1)^4 - 5(-1)^2 = 1 - 5 = -4 \0 & (0)^4 - 5(0)^2 = 0 \1 & (1)^4 - 5(1)^2 = 1 - 5 = -4 \3 & (3)^4 - 5(3)^2 = 81 - 45 = 36 \\hline\end{array}\]The table suggests that as \( x \) moves away from zero in either the positive or negative direction, \( f(x) \) increases, thus confirming the end behavior analysis.

Step 5: Summarize the End Behavior

Based on the leading term analysis and table values, the end behavior of \( f(x) = x^4 - 5x^2 \) is such that both ends of the graph rise upwards as \( x \to -\infty \) and \( x \to +\infty \).

Key Concepts

Polynomial function

A polynomial function is a mathematical expression composed of variables, coefficients, and the operations of addition, subtraction, and multiplication. It involves terms like \( a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \). Each term can have a variable raised to a non-negative integer exponent.

• Polynomials can have one or many terms, each with varying degrees.
• The expression \( f(x) = x^4 - 5x^2 \) is a polynomial function.
• It is a simple yet powerful form that allows easy computation and analysis.

Understanding polynomial functions helps predict values and behavior as inputs change. Their structured equations are straightforward for performing operations like addition or differentiation.

Leading term

The leading term of a polynomial is the term with the highest degree. It is crucial because it largely impacts the overall shape and end behavior of the polynomial's graph.

• For \( f(x) = x^4 - 5x^2 \), the leading term is \( x^4 \).
• It determines the polynomial's highest power and significantly affects the function's behavior when \( x \) approaches positive or negative infinity.

Identifying the leading term is the first step in understanding a polynomial's properties, such as its degree and how the graph behaves at extreme values of \( x \).

Degree of polynomial

The degree of a polynomial is the highest exponent of the variable \( x \) in its expression. It tells us how many times the function could potentially intersect a horizontal line, and it gives insight into the overall shape and possible peaks and troughs of the graph.

• For our function \( f(x) = x^4 - 5x^2 \), the degree is 4.
• This means it is an even degree, which influences the end behavior of the polynomial's graph.

The degree helps determine important characteristics, such as symmetry and the number of turns the graph might have. Even degrees signify symmetric end behavior, with both ends pointing in the same direction when the leading coefficient is positive.

Leading coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It plays a pivotal role in determining the direction of the graph's ends.

• In the polynomial \( f(x) = x^4 - 5x^2 \), the leading coefficient is 1.
• A positive leading coefficient (as in this case) indicates that the graph rises as \( x \) moves towards positive or negative infinity.

Understanding the leading coefficient helps predict the general shape and the orientation of the polynomial's graph. When combined with the degree, it forms a clear picture of how the function behaves at its ends.