Problem 30 - Precalculus : Building Concepts and Connections

Question

Determine the end behavior of the function. $$f(x)=3 x^{3}-4 x^{2}+5$$

Step-by-Step Solution

Answer

As \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity and as \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity.

Step 1: Identify Leading Term

The leading term of a polynomial function is the term with the highest degree. Here, the leading term is \(3x^3\). The coefficient of this term is 3, which is a positive number.

Step 2: Determine End Behavior

Since the degree of our leading term, \(3x^3\), is odd and its coefficient is positive, the end behavior of this function is: as \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity and as \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity. This is based on the end behavior pattern for polynomial functions.

Key Concepts

Leading Term

The leading term in a polynomial function is crucial as it dictates the overall behavior of the graph at its extreme ends. In a polynomial function, the leading term is identified by the variable with the highest exponent. For example, in the function given by the exercise, \( f(x) = 3x^3 - 4x^2 + 5 \), the leading term is \( 3x^3 \).

• The leading term contains the variable raised to the highest power.
• It's important because it primarily determines the function's end behavior, especially for very large or very small values of \( x \).

This leading term impacts the direction and steepness of the graph as values extend towards infinity, or negative infinity, since the other terms' influences diminish considerably as \( x \) grows larger.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables, multiplied by coefficients. A typical polynomial might look like \( a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \), where each \( a \) is a coefficient that can be any real number, and \( x \) is the variable.

• Each term in a polynomial function is a product of a coefficient and a non-negative integer power of \( x \).
• Polynomials are versatile and can describe a wide variety of curves or shapes on a graph.

Understanding polynomial functions is essential as they are one of the simplest forms used to model curves in calculus and algebra. The degree of the polynomial function, determined by the highest power of \( x \), is central to predicting the behavior and graph shape of the function.

Odd Degree Polynomials

A polynomial's degree is the highest power of the variable within the function. For polynomials of odd degree, their end behaviors are predictable based on the sign of the leading coefficient. If the leading coefficient is positive, as \( x \) approaches positive infinity, \( f(x) \) increases towards positive infinity and as \( x \) goes towards negative infinity, so does \( f(x) \). This consistent behavior happens because odd degree polynomials will always have an opposite end behavior on each side of the y-axis.

• Odd degree implies an odd number of roots or x-intercepts, though not all need to be real.
• The graph of an odd degree polynomial will always start in one vertical direction and end in the opposite.

For the function \( f(x) = 3x^3 - 4x^2 + 5 \), the leading term \( 3x^3 \) confirms that it is an odd-degree polynomial. Since the coefficient is positive, the end behavior matches the description: ascending to the right and descending to the left.