Problem 1 - Precalculus Mathematics for Calculus

Question

If the quantities \(x\) and \(y\) are related by the equation \(y=3 x\), then we say that \(y\) is _____ _____ to \(x\) and the constant of _____ is 3.

Step-by-Step Solution

Answer

y is directly proportional to x; constant of proportionality is 3.

Step 1: Understand the Relationship

In the given problem, the equation that relates the quantities is \(y = 3x\). This equation expresses a direct variation where \(y\) is dependent on \(x\).

Step 2: Identify the Type of Proportionality

The relationship \(y = 3x\) indicates that as \(x\) increases or decreases, \(y\) will increase or decrease proportionally. This type of relationship is known as "directly proportional." Therefore, \(y\) is directly proportional to \(x\).

Step 3: Determine the Constant of Proportionality

In the equation \(y = 3x\), the constant multiplier "3" is the constant of proportionality. It reflects how much \(y\) changes with a unit change in \(x\).

Key Concepts

Constant of Proportionality

When two quantities are related in a way that one quantity is a constant multiple of the other, we encounter what is known as the constant of proportionality. In simple terms, this constant tells us how one variable changes in relation to another.
For example, in the equation \( y = 3x \), the constant of proportionality is 3. This means that for every unit increase in \( x \), \( y \) increases by 3 units.

• If the constant of proportionality is greater than 1, the dependent variable increases faster than the independent variable.
• If it is less than 1 but greater than zero, the dependent variable increases at a slower rate than the independent variable.

This constant acts as a scaling factor between the two quantities, clearly defining their rate of change.

Linear Relationship

A linear relationship describes a connection between two variables that graph as a straight line on a coordinate plane. These relationships are governed by linear equations like \( y = mx + b \).
In our example of \( y = 3x \), the equation forms a straight line through the origin, representing a linear relationship with no y-intercept other than zero.

• The slope of a linear relationship, represented by \( m \) in the equation \( y = mx + b \), determines the steepness and direction of the line.
• In \( y = 3x \), the slope is 3, indicating that the line rises three units vertically for every unit it moves horizontally.

Linear relationships are foundational in math because they are simple yet widely applicable in representing constant rates of change.

Proportionality

Proportionality is a concept that refers to two quantities varying in such a way that they maintain a constant ratio or fraction. When we say two quantities are proportional, it means they increase or decrease in unison according to this set ratio.
In the equation \( y = 3x \), \( y \) is directly proportional to \( x \). This implies that as \( x \) grows, \( y \) grows three times as fast, adhering to the constant of proportionality, which is 3.

• Direct proportionality can be visually represented by a straight line that passes through the origin in a graph.
• It showcases a predictable and uniform rate of change between the two quantities.

Understanding proportionality helps in predicting how changes in one variable will affect another, forming a cornerstone in both basic mathematics and real-world applications.