Problem 1 - College Algebra with Corequisite Support

Question

What is true of the appearance of graphs that reflect a direct variation between two variables?

Step-by-Step Solution

Answer

Graphs of direct variation are straight lines through the origin.

Step 1: Understand Direct Variation

In a direct variation, two variables are related in such a way that the ratio of their corresponding values is constant. Mathematically, if variable \( y \) is directly proportional to variable \( x \), this relationship can be expressed as \( y = kx \), where \( k \) is a constant of proportionality.

Step 2: Identify the Graph Type

Graphs representing direct variation relationships are linear. This means they form a straight line when plotted on a coordinate plane because the equation \( y = kx \) has the form of a linear equation, \( y = mx + b \), where \( m \) is the slope, and \( b = 0 \).

Step 3: Analyze Line Characteristics

Since the direct variation equation is \( y = kx \), the graph will pass through the origin (0,0). This is a key characteristic because when \( x = 0 \), \( y = k \times 0 = 0 \). Thus, the line goes through the origin.

Step 4: Understand the Slope

The slope of the line is determined by the constant \( k \). If \( k \) is positive, the line will slope upwards, indicating a positive relationship between \( x \) and \( y \). Conversely, if \( k \) is negative, the line will slope downwards, showing a negative relationship.

Key Concepts

Linear Equations

Linear equations are the backbone of many algebraic concepts. They are called "linear" because they graph as straight lines on the coordinate plane. A general linear equation in two variables can be represented as \( y = mx + b \). Here, \( m \) stands for the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis.
Linear equations can describe a variety of real-world relationships.
In the context of direct variation, the linear equation simplifies to \( y = kx \), where \( b = 0 \). This signifies that the line passes through the origin.
This simple step from \( y = mx + b \) to \( y = kx \) makes direct variation equations a special subset of linear equations where there's no vertical offset.
Graphing a linear equation helps visualize the relationship between the two variables involved.
Recognizing the linear equation's characteristics can be very useful for understanding how variables interact and predict changes.

Constant of Proportionality

In the case of direct variation, the constant of proportionality is a key factor. This constant, represented by \( k \), defines the rate at which two variables change with respect to each other.
The mathematical expression for this is \( y = kx \). The constant of proportionality tells you how much one variable (\( y \)) changes when the other variable (\( x \)) changes.
If \( k = 2 \), for instance, it means that for every increase of 1 unit in \( x \), \( y \) will increase by 2 units.

• A larger absolute value of \( k \) implies a steeper line.
• A \( k \) of zero would mean no change in \( y \), resulting in a flat line, but in direct variation, \( k \) should not be zero.
A positive \( k \) results in an upward-sloping line, while a negative \( k \) yields a downward-sloping line.
This constant helps encapsulate the nature of the linear relationship, allowing for easy predictions and computations.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It's calculated as the ratio of the change in \( y \) (the rise) to the change in \( x \) (the run). In a direct variation equation \( y = kx \), the slope is directly represented by the constant \( k \).
The concept of slope is crucial for understanding linear relationships.

• A positive slope indicates an increasing line where the value of \( y \) rises as \( x \) increases.
• Conversely, a negative slope shows a decreasing line where \( y \) decreases as \( x \) increases.

In real-world terms, the slope can depict rates such as speed or cost per item, making it a valuable concept in various fields.
Recognizing the relationship between slope and the linear graph can make solving problems and predicting trends much clearer.