Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window. Zoom out far enough to see the right-hand and left-hand behavior of each graph. Do the graphs of \(f\) and \(g\) have the same right-hand and Ieft- hand behavior? Explain why or why not. \(f(x)=-\left(x^{4}-4 x^{3}+16 x\right), \quad g(x)=-x^{4}\)

Step-by-Step Solution

Answer

The functions do not have the same right-hand and left-hand behavior. This is due to the differences in their equations, where \(f(x)\) has terms such as \(4x^{3}\) and \(16x\) along with \(x^{4}\), whereas \(g(x)\) only has the \(x^{4}\) term.

Step 1: Graph the equation

Start off by graphing both functions on the same graph. Be sure to zoom out enough to observe the right-hand and left-hand behavior of each function.

Step 2: Observe the right-hand behavior

Observe the graphs as x approaches positive infinity (moving towards the right). Check if the outputs/values of \(f(x)\) and \(g(x)\) also tend to infinity, negative infinity, or a distinct constant.

Step 3: Observe the left-hand behavior

Observe the graphs as x approaches negative infinity (moving towards the left). Similarly, check if the outputs/values of \(f(x)\) and \(g(x)\) tend to infinity, negative infinity, or a distinct constant.

Step 4: Comparison

Compare the right-hand and left-hand behavior of the two given functions \(f(x)\) and \(g(x)\). Confirm if they exhibit the same behavior as x tends towards both positive and negative infinity.

Key Concepts

Right-hand Behavior

When analyzing the right-hand behavior of a function, we focus on what happens to the function's values as the input, or \(x\), becomes very large (approaches positive infinity). In our exercise, two functions \(f(x)\) and \(g(x)\) are being compared. Both functions have terms like \(-x^4\) indicating that as \(x\) increases, \(x^4\) will dominate the behavior of the equations.
Since the leading coefficient (the coefficient of \(x^4\)) is negative in both functions, this suggests that, for large values of \(x\), the output will approach negative infinity. This is typical behavior for a polynomial with an even degree and a negative leading coefficient. Hence, both \(f(x)\) and \(g(x)\) exhibit the same right-hand behavior, tending towards negative infinity as \(x\) approaches positive infinity.

Left-hand Behavior

The left-hand behavior looks at what happens as \(x\) approaches negative infinity. Similar to the right-hand behavior, we consider the dominant term \(-x^4\) which is present in both functions \(f(x)\) and \(g(x)\).
Since \(x^4\) retains its even nature, \((-x)^4 = x^4\), regardless of the negative \(x\), the outputs will once again be influenced primarily by the leading term. The negative in front indicates that as \(x\) decreases further, the function's values will go down towards negative infinity. Therefore, both functions \(f(x)\) and \(g(x)\) will similarly tend towards negative infinity as \(x\) approaches negative infinity. This means they share the same left-hand behavior.

Polynomial Functions

Polynomial functions are expressions formed using variables like \(x\) raised to whole number powers with coefficients. In this exercise, we deal with two polynomial functions: \(f(x) = -(x^4 - 4x^3 + 16x)\) and \(g(x) = -x^4\).
Key aspects of polynomials include their degree, leading coefficient, and the number of terms.

• The degree tells us how the ends of the graph behave. Here, both have a degree of 4, indicating even symmetry.
• The leading coefficient, especially when negative as it is here, tells us the direction in which the ends of the graph point—downwards towards negative infinity.
• The terms affect the graph's shape but do not change the end behavior, which is primarily determined by the leading term.

Understanding these characteristics helps in predicting and explaining the overall shape and behavior of polynomial graphs, especially on advanced topics like end behavior and graph intercepts.