Problem 41 - Algebra and Trigonometry

Question

\(41-46\) . Determine the end behavior of \(P\) . Compare the graphs of \(P\) and \(Q\) in large and small viewing rectangles, as in Example 3\((\mathrm{b})\) . $$ P(x)=3 x^{3}-x^{2}+5 x+1 ; \quad Q(x)=3 x^{3} $$

Step-by-Step Solution

Answer

Both \( P(x) \) and \( Q(x) \) rise to the right and fall to the left, but \( P(x) \) has more complex behavior in small view.

Step 1: Analyze the Degree and Coefficients of P

The polynomial function \( P(x) = 3x^3 - x^2 + 5x + 1 \) is a cubic polynomial (degree 3) with the leading term \( 3x^3 \). Since the leading term determines the end behavior, \( P(x) \) will have the same end behavior as a basic cubic function, which is determined by the coefficient of the leading term, in this case, 3.

Step 2: Determine End Behavior of P

For the polynomial \( P(x) = 3x^3 - x^2 + 5x + 1 \), as \( x \to \infty \), \( P(x) \to \infty \) because the leading term \( 3x^3 \) dominates and its coefficient is positive. As \( x \to -fty \), \( P(x) \to -fty \).

Step 3: Analyze Function Q

The function \( Q(x) = 3x^3 \) is a simple cubic polynomial with the same leading term \( 3x^3 \) as \( P(x) \). Its end behavior will also be \( Q(x) \to fty \) as \( x \to fty \) and \( Q(x) \to -fty \) as \( x \to -fty \).

Step 4: Compare Graphs in Large Viewing Rectangle

In a large viewing rectangle, both \( P(x) \) and \( Q(x) \) will display similar end behavior due to their cubic leading term. Thus, both graphs will rise to the right and fall to the left, mirroring each other at extremes.

Step 5: Compare Graphs in Small Viewing Rectangle

In a small viewing rectangle, the additional terms in \( P(x) = 3x^3 - x^2 + 5x + 1 \) will affect the shape of the graph, making it more complex compared to \( Q(x) = 3x^3 \), which remains a simple cubic curve without additional inflections or deviations.

Key Concepts

Cubic Functions

Cubic functions are a type of polynomial function where the highest degree of the variable is three. A typical cubic function can be written as \( ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a eq 0 \).
They are generally known for having an "S"-shaped curve. This is due to the nature of cubic functions having one or two turning points which results in one or two humps on their graph. These functions are continuous and smooth, not having any breaks or sharp corners.
When evaluating cubic functions like \( P(x) = 3x^3 - x^2 + 5x + 1 \), we look at how these polynomials behave as \( x \) values become very large or very small. This behavior is especially crucial when exploring how the graphs change across different zoom levels in graphing software or calculators.

Leading Term

The leading term of a polynomial is the term with the highest power of \( x \). It plays a crucial role in determining the end behavior of the polynomial because it grows the fastest as \( x \) increases or decreases in magnitude.
In the polynomial \( P(x) = 3x^3 - x^2 + 5x + 1 \), the leading term is \( 3x^3 \). This means that as \( x \to \infty \), the leading term \( 3x^3 \) dictates that \( P(x) \to \infty \) because the coefficient is positive. Conversely, as \( x \to -\infty \), \( 3x^3 \to -\infty \), making \( P(x) \to -\infty \).
Understanding the leading term is essential when comparing functions, such as \( P(x) \) and \( Q(x) = 3x^3 \). Both have the same leading term, so at extreme values, their behavior aligns closely.

Graph Comparison

When comparing graphs of polynomials, it helps to change the scale or size of the viewing window. This can showcase different features of the polynomial.

• **Large Viewing Rectangle**: In a large viewing rectangle, the end behavior dictated by the leading term becomes prominent. For both \( P(x) \) and \( Q(x) \), the graphs rise to the right and fall to the left due to their shared leading term \( 3x^3 \).
• **Small Viewing Rectangle**: In contrast, a small viewing rectangle reveals additional details in \( P(x) \) that \( Q(x) = 3x^3 \) does not have. The terms \(-x^2 + 5x + 1\) influence the graph, introducing more complexity such as humps or dips not found in \( Q(x) \).

These comparisons help students grasp how polynomial terms beyond the leading one can alter the overall shape and behavior of the graph.

Polynomial Degree

The degree of a polynomial is determined by the highest power of the variable present in the function. It tells us about the possible number of roots, turns, and the overall structure of the graph.
In the case of cubic polynomials, like \( P(x) = 3x^3 - x^2 + 5x + 1 \), the degree is 3. This degree indicates that the function can have up to three zeros (or roots) and potentially two turning points.
Understanding the degree helps in predicting the graph's complexity. For example, a higher degree often means more oscillations or inflection points. This is why \( P(x) \) appears more intricate than \( Q(x) = 3x^3 \) in smaller viewing windows, as its additional non-leading terms contribute to the polynomial's overall degree and complexity.