Problem 31 - Precalculus: Functions and Graphs

Question

Exer. 31-34: Examine the expression for the given set of data points of the form \((x, y) .\) Find the constant of variation and a formula that describes how \(y\) varles with respect to \(x .\) $$\begin{array}{c} y / x ;\\{(0.6,0.72),(1.2,1.44),(4.2,5.04),(7.1,8.52), \\ (9.3,11.16)\\} \end{array}$$

Step-by-Step Solution

Answer

The direct variation formula is \(y = 1.2x\).

Step 1: Understand Direct Variation

A direct variation between two variables \(x\) and \(y\) is described by the equation \(y = kx\), where \(k\) is the constant of variation. Our objective is to find \(k\) based on the given data points.

Step 2: Calculate the Constant of Variation \(k\)

For each data point \((x, y)\), calculate \(k = \frac{y}{x}\). This will give us the constant of variation if \(y\) varies directly with \(x\).

Step 3: Compute \(k\) for Each Data Point

Using the formula \(k = \frac{y}{x}\), compute \(k\) for all given data points:- For \((0.6, 0.72)\), \(k = \frac{0.72}{0.6} = 1.2\)- For \((1.2, 1.44)\), \(k = \frac{1.44}{1.2} = 1.2\)- For \((4.2, 5.04)\), \(k = \frac{5.04}{4.2} = 1.2\)- For \((7.1, 8.52)\), \(k = \frac{8.52}{7.1} = 1.2\)- For \((9.3, 11.16)\), \(k = \frac{11.16}{9.3} = 1.2\)

Step 4: Analyze the Results

All calculations yield the same constant \(k = 1.2\). This indicates that \(y\) varies directly with \(x\) and the relationship is consistent across all data points.

Step 5: Write the Formula

Since \(k = 1.2\) for all data points, the direct variation formula that describes how \(y\) varies with respect to \(x\) is \(y = 1.2x\).

Key Concepts

Constant of Variation

The constant of variation is a key component in understanding how two variables relate to one another in a direct variation format. When variables show a direct variation, it means that as one variable increases or decreases, the other does so in a proportional manner. This relationship is captured by a constant value, which remains the same across different data pairs. In the equation of direct variation, expressed as \( y = kx \), the constant \( k \) represents how much \( y \) changes for every unit change in \( x \). For the given data points, regardless of which pair you choose, \( k \) was the same, \( 1.2 \), indicating a consistent rate of change. Consistent constants confirm a stable relationship, defining how the variables move in harmony.

Data Points

Data points are specific pairs of numerical values which represent the relationship between two variables. They serve as evidence to analyze and confirm hypotheses about how variables are related. In this exercise, each data point consists of an \( x \) and \( y \) value, such as \((0.6, 0.72)\) and \((4.2, 5.04)\). By examining these points, you can derive meaningful conclusions such as the presence of a direct variation.

• A data set that fits a direct variation will show a consistent ratio \( \frac{y}{x} \) across all points.
• The uniform value of \( k = 1.2 \) across all provided data points reinforces that the variables exhibit direct variation.

Recognizing patterns in data points is crucial in identifying trends and establishing formulas that can predict or describe behavior in similar scenarios.

Mathematical Formula

A mathematical formula provides a concise way to express relationships between variables. In the context of direct variation, the formula \( y = kx \) is used. This formula tells us that \( y \), the dependent variable, scales directly with \( x \), the independent variable, by the constant \( k \). The formula is powerful because:

• It is derived from the observed ratio of \( y \) to \( x \) which remains consistent across all data points.
• It allows you to predict the value of \( y \) for any given \( x \) by simply multiplying \( x \) by \( k \).
• Having a precise mathematical relationship simplifies complex real-world phenomena into manageable calculations.

In this specific instance, the formula \( y = 1.2x \) effectively summarizes the relationship giving us the ability to swiftly calculate \( y \) when \( x \) is known.

Linear Relationship

A linear relationship is characterized by a constant rate of change, which is depicted as a straight line when graphed. In the case of direct variation, this linear relationship can be seen through the equation \( y = kx \). This indicates that:

• Both variables increase or decrease at a constant rate represented by the constant of variation, \( k \).
• The graph of such a relationship is a line passing through the origin, illustrating equal proportions of \( y \) to \( x \).
• This is why in our data points, the ratio \( \frac{y}{x} = 1.2 \) remains consistent, affirming the linear nature of the relationship.

Understanding linear relationships in direct variation situations helps simplify complex data analysis and predicts behaviors in diverse fields such as physics, economics, and statistics.