Problem 52 - College Algebra

Question

The common cold is caused by a rhinovirus. After \(x\) days of invasion by the viral particles, the number of particles in our bodies, \(f(x),\) in billions, can be modeled by the polynomial function $$ f(x)=-0.75 x^{4}+3 x^{3}+5 $$ Use the Leading Coefficient Test to determine the graphs end behavior to the right. What does this mean about the number of viral particles in our bodies over time?

Step-by-Step Solution

Answer

Using the Leading Coefficient Test, it can be derived that as the number of days increases, the number of viral particles in the body decreases.

Step 1: Identify the highest power and leading coefficient

The function provided is \(f(x)=-0.75x^{4}+3x^{3}+5\). Here, 4 is the highest power and -0.75 is its coefficient, hence called the leading coefficient.

Step 2: Applying the Leading Coefficient Test

The Leading Coefficient Test tells us that if the degree of the polynomial is even, and the leading coefficient is positive, the graph rises to the right and left. If the leading coefficient is negative, the graph falls to the right and left. In our case, we have an even degree (4) and a negative leading coefficient (-0.75), hence the function decreases as x increases.

Step 3: Interpret the result

The graph’s end behavior suggests that as the number of days (x) goes to infinity, the number of virus particles in the body, quantified by \(f(x)\), decreases. Hence over time, the number of viral particles in the body decreases.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a handy tool for predicting how a polynomial function behaves at the far ends of the graph, particularly as \( x \) approaches positive or negative infinity. In simpler terms, it helps us understand the "end behavior" of the graph.
Here's how it works:

• Identify the leading coefficient (the number in front of the highest degree term).
• Check the degree of the polynomial (even or odd).
• Combine this information to predict the end behavior.

For an even degree polynomial:

• If the leading coefficient is positive, the graph rises on both sides.
• If it's negative, it falls on both sides.

Our example function, \( f(x) = -0.75x^{4} + 3x^{3} + 5 \), has an even degree with a negative leading coefficient, indicating the graph falls as \( x \) increases.
This means the viral count is projected to decrease over the days.

End Behavior

End behavior describes how the graph behaves as the input \( x \) moves to positive or negative infinity. It's a crucial aspect of analyzing polynomial functions, as it helps anticipate long-term trends.
When considering end behavior, observe a few key details:

• The sign and degree of the leading term heavily influence the graph behavior.
• For even degree polynomials with a negative leading coefficient, both ends of the graph will move downwards, or fall.
• This is in contrast with odd degree polynomials, where the sides will move in opposite directions.

Looking at our function, \( f(x) \), both ends of the graph trend down due to the even degree (4) and negative coefficient (-0.75). This suggests fewer virus particles as time stretches on.

Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial, which plays a significant role in determining the shape of a graph. For our function \( f(x) = -0.75x^{4} + 3x^{3} + 5 \), the degree is 4.
Why is the degree important? Here are a few main points:

• It determines the maximum number of turning points the graph can have.
• For an even degree, the ends of the graph tend to move in the same direction (either up or down, depending on the leading coefficient).
• The larger the degree, the more complex the graph's shape can become.

In practical terms, a degree of 4 implies that the graph may have up to three turning points (one less than the degree) and the end behavior can be predicted using the leading coefficient.
Thus, understanding the degree helps us interpret how a function behaves over time, seeing past just the immediate rise and fall of the graph.