Problem 69 - Algebra and Trigonometry

Question

\(69-74\) . Graph the family of polynomials in the same viewing rectangle, using the given values of \(c\) . Explain how changing the value of \(c\) affects the graph. $$ P(x)=c x^{3} ; \quad c=1,2,5, \frac{1}{2} $$

Step-by-Step Solution

Answer

The value of \( c \) changes the steepness of the polynomial graph; higher \( c \) values make it steeper.

Step 1: Understanding the Problem

We need to graph the polynomial function \( P(x) = c x^3 \) for different values of \( c \) including 1, 2, 5, and \( \frac{1}{2} \). Our task is to identify how different values of \( c \) affect the graph.

Step 1: Analyzing the Function with c = 1

When \( c = 1 \), the polynomial becomes \( P(x) = x^3 \). This is a basic cubic function passing through the origin with a standard cubic shape: increasing for \( x > 0 \) and decreasing for \( x < 0 \).

Step 2: Analyzing the Function with c = 2

For \( c = 2 \), the polynomial is \( P(x) = 2x^3 \). This function stretches the basic cubic graph vertically by a factor of 2. As a result, it rises more steeply for positive \( x \) and falls more steeply for negative \( x \) compared to when \( c = 1 \).

Step 3: Analyzing the Function with c = 5

For \( c = 5 \), the polynomial becomes \( P(x) = 5x^3 \). This further stretches the graph vertically, making it even steeper than when \( c = 2 \). The changes in \( y \) values for given \( x \) values are five times that of the basic cubic function.

Step 4: Analyzing the Function with c = 0.5

When \( c = \frac{1}{2} \), the polynomial is \( P(x) = \frac{1}{2}x^3 \). This reduces the steepness of the graph compared to the basic cubic function, making it half as steep as when \( c = 1 \). It represents a compressed version of the cubic graph.

Step 5: Graphical Comparison

When graphing all these functions within the same viewing rectangle, you'll notice that changing \( c \) modifies the steepness or slope of the cubic graph. Larger values of \( c \) stretch the graph vertically, making it steeper, while smaller values like \( \frac{1}{2} \) compress it, decreasing the steepness.

Key Concepts

Cubic Functions

Cubic functions are polynomial functions given by the general form \( f(x) = ax^3 + bx^2 + cx + d \). A cubic function usually has its highest degree as 3 and is thus called a third-degree polynomial. These functions are characterized by their distinctive shape, which often includes one or two bends called inflection points. For the function \( P(x) = x^3 \), it is a simple cubic function that passes through the origin (0,0) and is symmetric about this point. It increases steeply where \( x > 0 \) and decreases where \( x < 0 \). This simple cubic graph has no turning points as its smooth curve transitions smoothly from negative infinity to positive infinity through the origin.

Graph Transformations

Graph transformations involve systematically altering a function's appearance on a coordinate plane. The basic transformations include shifting, reflecting, stretching, and compressing the graph. When we alter the function \( P(x) = c x^3 \) by changing \( c \), we are applying vertical transformations. These affect how the graph moves along the y-axis without changing its shape completely.

• A positive \( c \) keeps the graph in the same direction (increases), while a negative \( c \) would reflect it.
• When \( c \) is more than 1, the graph stretches, and it compresses when between 0 and 1.

This specific transformation retains the characteristic shape of a cubic function but may make it appear steeper or flatter depending on the value of \( c \).

Vertical Stretching and Compression

Vertical stretching and compression directly influence the steepness of a graph. Vertical Stretching: This occurs when the constant \( c \) in \( P(x) = c x^3 \) is greater than 1. For instance, \( c = 2 \) and \( c = 5 \) scale the graph vertically, making the slope steeper and the points rise or drop more sharply than the basic curve \( P(x) = x^3 \). With stretching, the output or \( y \)-values become a multiple of the original function.Vertical Compression: This effect takes place when \( 0 < c < 1 \). As seen with \( c = \frac{1}{2} \), it compresses the cubic graph, reducing its steepness. The effect is such that for any chosen \( x \)-value, the associated \( y \) is only a fraction of that in the original \( P(x) = x^3 \). This reduced slope is akin to flattening the curve, expanding its spread along the x-axis without altering its base form.