Problem 48 - Precalculus : Building Concepts and Connections

Question

Find a function of the form \(y=c x^{k}\) that has the same end behavior as the given function. Confirm your \(\mathrm{re}\) sults with a graphing utility. $$g(x)=-3.6 x^{4}+4 x^{2}+x-20$$

Step-by-Step Solution

Answer

The function \(y=-3.6 x^{4}\) has the same end behavior as the given function \(g(x)=-3.6 x^{4}+4 x^{2}+x-20\).

Step 1: Identify the leading term of \(g(x)\)

In the polynomial \(g(x)=-3.6 x^{4}+4 x^{2}+x-20\), the leading term is the term with the highest degree. Here, the highest degree is 4, so the leading term is \(-3.6x^{4}\).

Step 2: Create the function with the same end behavior

The function with the same end behavior should share the same degree and coefficient as the leading term. Hence, the function is \(y= -3.6 x^{4}\).

Step 3: Confirming the result with a graphing utility

By graphing both functions \(g(x)=-3.6 x^{4}+4 x^{2}+x-20\) and \(y=-3.6 x^{4}\), one can see that both the functions have the same end behavior, both tending towards \(-\infty\) when \(x \rightarrow \pm \infty\). Hence, the solution is confirmed.

Key Concepts

Leading Term

The leading term in a polynomial function plays a key role in determining its end behavior. In the polynomial expression, it is the term with the highest power of the variable. For example, in the polynomial \( g(x) = -3.6x^4 + 4x^2 + x - 20 \), the leading term is \(-3.6x^4\), because it is associated with the highest power, which is 4.

Identifying the leading term is essential when you need to assess the behavior of the polynomial as \( x \) approaches infinity or negative infinity. Observing the leading term helps predict whether the function will tend toward positive or negative infinity. In this particular polynomial, the negative coefficient \(-3.6\) ensures that as \( x \) becomes very large or very small, the function \( g(x) \) will tend toward \(-\infty\).

So, it's not just about spotting the highest power, but also considering its coefficient, which tells you whether the end behavior is going to be reaching up or down toward infinity.

Degree of Polynomial

The degree of a polynomial is an important characteristic that tells you about its overall shape and complexity. The degree is simply the highest power of the variable in the polynomial. In our example \( g(x) = -3.6x^4 + 4x^2 + x - 20 \), the degree is determined by the leading term \(-3.6x^4\), so the degree is 4.

A polynomial’s degree indicates the maximum number of turns the graph can have and predicts the end behavior of the function. Generally, the end behavior depends largely on whether the degree is even or odd:

• If even, both ends of the function's graph will point in the same direction.
• If odd, the ends will point in opposite directions.

For a degree of 4, which is even, both ends of the graph will point in the same direction, influenced by the leading coefficient, which in this case is negative \(-3.6\). This ensures both ends of the graph are pointing downwards toward matching \(-\infty\).

Graphing Utility

A graphing utility is a helpful tool for visualizing mathematical functions. It can be anything from a graphing calculator to software like Desmos or GeoGebra. These tools allow you to input functions and instantly see their graphs, which is extremely helpful for confirming theoretical findings about a function's end behavior and other characteristics.

When dealing with polynomials, using a graphing utility can show the practical shapes and trends of the graph that might be difficult to piece together just from inspecting the algebraic expression. In the case of \( g(x) = -3.6x^4 + 4x^2 + x - 20 \) and \( y = -3.6x^4 \), graphing confirms that both functions exhibit the same end behavior. Both tend toward \(-\infty\) as \( x \) goes towards positive or negative infinity.

One advantage of a graphing utility is its capability to display how small terms in the polynomial, like \(4x^2\) or \(x\), become negligible when \( x \) is very large. Thus, the graph closely follows that of the leading term for large \( x \) values.

Polynomial Function

A polynomial function is a mathematical expression made up of variables raised to whole number powers and coefficients. It generally has the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( n \) is a non-negative integer. Each term in this expression is a product of a constant coefficient \( a_i \) and a power of \( x \).

Polynomials are very important in mathematics due to their simple structure and wide applicability.

• Easily integrated and differentiated.
• Graphing provides insights into real-world physical phenomena.

In our example, \( g(x) = -3.6x^4 + 4x^2 + x - 20 \), each coefficient plays a role in altering the graph's shape, thus manipulating intercepts and turning points.

Understanding polynomial functions is essential for delving into more complex areas of math, and it serves as the groundwork for studying calculus and applied mathematics. Functions like \( y = -3.6x^4 \) can be simplified to focus on the dominant behavior of \( g(x) \) as shown in exercises on end behaviors.