Problem 47 - College Algebra

Question

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 of the function is: as \( x \to \pm \infty \), \( f(x) \to +\infty \).

Step 1: Identify the leading term of the function

The function given is \( f(x) = x^4 - 5x^2 \). The leading term is the one with the highest power of \( x \). In this case, the leading term is \( x^4 \). This term will primarily dictate the end behavior of the function because it increases at a faster rate than the other terms as \( x \) becomes very large or very small.

Step 2: Determine the end behavior using the leading term

The end behavior of a polynomial function is determined by the leading term's degree and the sign of its coefficient. Here, the leading term is \( x^4 \), which is a positive coefficient and even degree. For even-degree polynomials with a positive leading coefficient, as \( x \to +\infty \), \( f(x) \to +\infty \), and as \( x \to -\infty \), \( f(x) \to +\infty \).

Step 3: Explain using a table

Create a table of \( x \) values to observe the function's behavior as it approaches infinity in both directions. Choose values of \( x \) that are large and extreme:\[\begin{array}{|c|c|}\hlinex & f(x) = x^4 - 5x^2 \\hline10 & 10^4 - 5(10^2) = 10000 - 500 = 9500 \-10 & (-10)^4 - 5(-10)^2 = 10000 - 500 = 9500 \20 & 20^4 - 5(20^2) = 160000 - 2000 = 158000 \-20 & (-20)^4 - 5(-20)^2 = 160000 - 2000 = 158000 \\ldots & \ldots \\hline\end{array}\]Notice that as \( |x| \) increases, \( f(x) \) becomes very large and positive, confirming the end behavior.

Key Concepts

Leading Term

One of the key elements in understanding the end behavior of a polynomial function is identifying the leading term. The leading term is the term with the highest degree, or most precisely, the highest power of the variable \( x \). In the polynomial function \( f(x) = x^4 - 5x^2 \), the leading term is \( x^4 \). This term overshadows all others as \( x \) becomes very large, either positively or negatively. This is because higher degree terms grow much faster than terms with lower degrees.
When evaluating polynomial functions for end behavior, always locate and focus on this term. Since it indicates how quickly it will grow, it ultimately dictates how the function behaves as \( x \) moves towards infinity or negative infinity.

Degree of Polynomial

The degree of a polynomial is crucial as it provides insight into the general shape and behavior of the polynomial graph. The degree is determined by the highest exponent in the polynomial. For the function \( f(x) = x^4 - 5x^2 \), the degree is 4, meaning it is a fourth-degree polynomial.
All polynomials of the same degree share certain key characteristics, especially regarding end behavior. A higher degree polynomial like this typically has a more complex structure than lower degree polynomials. The fourth degree indicates that the graph will have certain sets of ups and downs and can intersect the x-axis at most four times, though it is the leading term that is the most instructive regarding end behavior.

Even Degree Polynomial

Polynomials can be either even or odd in degree, and this aspect significantly influences their end behavior. In the given function \( f(x) = x^4 - 5x^2 \), the degree of 4 is an even number. An even degree polynomial generally means that the ends of the graph will approach infinity in the same direction. Since the coefficient of the leading term \( x^4 \) is positive, this function's ends both rise upwards.

• As \( x \to +\infty \), \( f(x) \to +\infty \).
• As \( x \to -\infty \), \( f(x) \to +\infty \).

This behavior is indicative of all even-degree polynomials with positive leading coefficients; both ends of the graph head upwards. Understanding this pattern can help you predict the end behavior simply by identifying these polynomial characteristics.

Table of Values

Creating a table of values is a straightforward way to confirm and visualize a polynomial's end behavior. By selecting a range of \( x \) values, especially values that are quite large or quite large in their negative form, you can observe how the function behaves.
For the function \( f(x) = x^4 - 5x^2 \), testing with values such as 10, -10, 20, and -20 can demonstrate the graph's trend as anticipated by its even degree and positive leading term. For instance, plugging \( x = 20 \) into the function yields:

• \( f(20) = 20^4 - 5(20)^2 = 158000 \)
• \( f(-20) = (-20)^4 - 5(-20)^2 = 158000 \)

Each of these calculations confirms that as \( |x| \) increases, the \( f(x) \) values become large and positive, demonstrating the polynomial's end behavior clearly. Tables like these allow for a tangible and numerical verification of theoretical conclusions.