Problem 84 - Precalculus with Limits

Question

In Exercises 83 and 84, use a graphing utility to graph \( f \) and \( g \) in the same viewing window.What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \( g \) in standard form. \( f(x) = -x^4 + 4x^2 - 1, g(x) = f\left(x - 3\right) \)

Step-by-Step Solution

Answer

The graph of function \( g \) is a rightward shift of the graph of function \( f \) by 3 units, and the standard form of \( g(x) \) is \( g(x) = -x^4 +12x^3 - 36x^2 +36x +22 \)

Step 1: Graph the functions

Use a graphing tool to graph the two functions \( f(x) = -x^4 + 4x^2 - 1 \) and \( g(x) = f\left(x - 3\right) \) in the same viewing window. Compare the two graphs to understand their relationship.

Step 2: Relationship between the graphs

From the graph, we can observe that the graph of function \( g \) is a rightward shift of the graph of function \( f \) by 3 units. This is because \( g(x) \) is defined as \( f(x-3) \), which means every x-value in \( f \) is replaced with \( x-3 \) in \( g \), resulting in a shift in the graph.

Step 3: Use the Binomial Theorem to write \( g(x) \)

The Binomial Theorem can be used to expand \( g(x) = f(x-3) \), where \( f(x) = -x^4 + 4x^2 - 1 \). After substituting \( x-3 \) into the equation of \( f \), we get \( g(x) = -(x-3)^4 + 4*(x-3)^2 -1 \). Expanding out, we find that \( g(x) = -x^4 +12x^3 - 36x^2 +36x +22 \). Here, the Binomial theorem has been used to transform the \( (x-3)^4 \) and \( (x-3)^2 \) into the standard polynomial form.

Key Concepts

Graphing Utility

Graphing utilities are essential tools for visualizing the behavior of functions and comparing relationships between them. For instance, the exercise involves graphing two functions, f(x) = -x^4 + 4x^2 - 1 and g(x) = f(x-3), within the same viewing window. By utilizing a graphing utility, students can observe that g(x) is a transformation of f(x), specifically a horizontal shift to the right controlled by the change in the variable x.

Such tools offer a dynamic approach to understanding mathematical concepts as they allow for immediate visual interpretation, helping to create a mental model of the function's shape, points of intersection, and overall structure. The graphing utility is a bridge between abstract algebraic expressions and their geometric representation, which solidifies comprehension and gives abstract concepts a tangible form.

Binomial Theorem

The Binomial Theorem is a powerful principle in algebra used to expand expressions that are raised to a power. In the provided exercise, the theorem assists in rewriting the function g(x) = f(x-3) into standard polynomial form. Given a binomial (a+b)^n, where n is a non-negative integer, the Binomial Theorem defines how to expand it into the sum of terms of the form a^{n-k}b^k, where the coefficient of each term is a specific binomial coefficient.

This theorem simplifies complex polynomial multiplication and is essential for understanding how algebraic operations alter the function's structure. When applied to the function g(x), the Binomial Theorem unveils the adjusted coefficients and provides insight into how transformations affect polynomial expressions.

Polynomial Function in Standard Form

A polynomial function in standard form is an expression where terms are ordered from highest to lowest degree. The initial function f(x) = -x^4 + 4x^2 - 1 is already presented in standard form. The transformation of this function into g(x) involves writing g(x) = -(x-3)^4 + 4*(x-3)^2 - 1 in standard form, which is achieved by applying the Binomial Theorem.

Writing polynomials in standard form is crucial as it offers a direct way to identify the degree of the polynomial and helps to determine end behavior, intercepts, and overall shape of the graph. It's a foundational skill in algebra that aids in the analysis, comparison, and composition of polynomial functions.

Transformation of Functions

Transformation of functions involves altering a parent function in order to shift, stretch, compress, or reflect its graph. In this exercise, we examine a horizontal shift, which is a type of transformation. Specifically, the graph of g(x) is created by shifting the graph of f(x) horizontally by 3 units to the right.

Understanding function transformations is key in mathematical analysis since it facilitates the study of how various alterations to the algebraic equation translate to movements and changes in the graph. This concept not only reinforces the connection between algebra and geometry but also empowers students to manipulate and control functions to suit specific needs, such as modeling real-world scenarios.