Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=(x-1)(x-2)(3-x) $$

Step-by-Step Solution

Answer

The function falls to the right and rises to the left.

Step 1: Identify the Leading Term

Multiply the first terms of each factor: \((x)(x)(-x)\). This simplifies to \(-x^3\), which determines the end behavior of the polynomial function.

Step 2: Determine the Degree and Leading Coefficient

The degree of the polynomial is 3, because \(-x^3\) is the term with the highest exponent. The leading coefficient is -1, from the term \(-x^3\).

Step 3: Analyze End Behavior Based on Degree and Coefficient

For - An odd degree (3), which indicates different end behaviors at positive and negative infinities. - A negative leading coefficient (-1), indicating that the graph falls to the right and rises to the left.

Step 4: Construct a Table of End Behavior

Create a table indicating behavior as \(x\) approaches positive and negative infinity:\[\begin{array}{|c|c|}\hlinex \to +\infty & f(x) \to -\infty \\hlinex \to -\infty & f(x) \to +\infty \\hline\end{array}\]This confirms that as \(x\) increases to the right, \(f(x)\) decreases, and as \(x\) decreases to the left, \(f(x)\) increases.

Key Concepts

Degree of a Polynomial

When we talk about the degree of a polynomial, we mean the highest power of the variable in the polynomial expression. The degree is a key factor in determining the end behavior of a polynomial function.
In the example provided, the degree of the polynomial is calculated by multiplying the first terms of the factors: \((x)(x)(-x)\), resulting in \(-x^3\).
This tells us that our polynomial is a cubic polynomial, meaning it has a degree of 3. Higher-degree polynomials can become quite complex, but their end behavior can still be understood fairly easily by focusing on their degree.

Leading Coefficient

The leading coefficient is incredibly important in understanding polynomial functions. It refers to the coefficient of the term with the highest degree. Essentially, it is the number in front of the variable raised to the highest power.
In our example, the leading term is \(-x^3\), and the leading coefficient is \(-1\).
The leading coefficient can inform us about the direction the graph will take. A positive leading coefficient typically means that as \(x\) moves towards positive infinity, the function will also rise. This is reversed when the leading coefficient is negative.

Odd Degree Polynomials

Odd degree polynomials have distinctive end behaviors on either side of the graph. Because they are not symmetric, the direction the graph takes as \(x\) approaches infinity is different from the direction as \(x\) goes towards negative infinity.
Our polynomial is of degree 3, which is odd, so we can expect that as \(x\) approaches \(+\infty\), the function will perform differently compared to as \(x\) approaches \(-\infty\).
For the polynomial given \(f(x)=(x-1)(x-2)(3-x)\), the odd degree tells us to look for these opposing behaviors at the extremes of the x-axis.

Negative Leading Coefficient

Having a negative leading coefficient significantly affects the shape and direction of a polynomial function's graph. It "flips" the graph over its typical behavior when the leading coefficient is positive.
This flipping means that if a normal polynomial rises to the right, with a negative leading coefficient, it will fall to the right instead. Similarly, if it falls to the left, it will rise to the left.
In our polynomial \(-x^3\), the negative leading coefficient \(-1\) means that, at x-positive infinity, the function \(f(x)\) heads towards negative infinity. Conversely, as \(x\) approaches negative infinity, \(f(x)\) will go towards positive infinity.

• This is a convenient way to predict graph behaviors without drawing them.
• It's essential for solving functional equations or analyzing graphs visually.