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Magazine Chapter 3: Problem 170
For the following exercises, determine the end behavior of the functions. $$f(x)=x^{3}$$
Short Answer
Expert verified
As \(x \to +\infty\), \(f(x) \to +\infty\); as \(x \to -\infty\), \(f(x) \to -\infty\).
Step by step solution
01
Analyze the Highest Degree Term
In the polynomial function \(f(x) = x^3\), the highest degree term is \(x^3\). Since it is a cubic term, it will determine the overall end behavior of the function.
02
Identify the Leading Coefficient
The leading coefficient of the term \(x^3\) is 1, which is a positive number. The sign of this coefficient will affect the direction of the ends of the function.
03
Determine the End Behavior Based on Degree and Coefficient
A positive cubic function, like \(f(x) = x^3\), will behave as follows: as \(x \to +\infty\), \(f(x) \to +\infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\). This is because cubic functions with positive coefficients increase from the lower left to the upper right on a graph.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. Each term consists of a coefficient and a variable raised to a non-negative integer exponent.
- The general form of a polynomial function in one variable is \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0 \), where \( a_n, a_{n-1}, \,..., a_1, \) and \( a_0 \) are constants.
- Polynomials are named according to their degree. For instance, a function is quadratic if it involves \( x^2 \), cubic if \( x^3 \), quartic if \( x^4 \), and so on.
- The simplicity and versatility of polynomial functions make them a fundamental concept in algebra. They are easy to graph and solve, displaying clear patterns.
Cubic Functions
Cubic functions form a specialized subset of polynomial functions. They are defined by having their highest exponent as 3, giving them a distinctive shape and behavior when graphed.
These functions can be written in the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a eq 0 \). Here's more about cubic functions:
These functions can be written in the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a eq 0 \). Here's more about cubic functions:
- Cubic graphs typically cross the x-axis up to three times, corresponding to their possible real roots.
- These functions have one inflection point, where the graph changes concavity.
- They exhibit specific end behaviors, influenced by the sign and value of the leading coefficient \( a \).
Leading Coefficient
The leading coefficient in any polynomial function is the coefficient of the term with the highest degree. It plays a crucial role in defining both the shape and the direction of the polynomial's graph. In \( f(x) = x^3 \), the leading coefficient is 1. Let's explore why this is important:
- The leading coefficient affects the graph's end behavior. A positive leading coefficient implies that the polynomial's graph will extend upwards on the right end, while a negative coefficient will direct it downward.
- In cubic functions, like \( f(x) = x^3 \), a positive leading coefficient results in a graph that moves up from the lower left to the upper right.
- Overall magnitude of the leading coefficient affects steepness or wideness of the graph arc.
Degree of the Polynomial
The degree of a polynomial is the highest power of the variable in the function. This important characteristic indicates a few things about the polynomial's properties. In the exercise with \( f(x) = x^3 \), the function's degree is 3, making it a cubic polynomial. Here's what the degree tells you:
- The degree determines the basic shape and number of roots of the polynomial. A degree of 3 suggests up to three real roots and two turning points.
- It also conveys the end behavior. For odd-degree polynomials like cubic ones, end behaviors are opposite as \( x \to +\infty \) and \( x \to -\infty \).
- Higher degree polynomials become more complex and nuanced in their graph structure.
