A direct proportion is a relationship between two variables where as one variable increases, the other variable increases at a constant rate. The variables move in the same direction, and the ratio between them remains constant.
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In a directly proportional relationship, the graph of the two variables is a straight line passing through the origin.
The equation for a directly proportional relationship is $y = kx$, where $k$ is the constant of proportionality.
Directly proportional relationships are often seen in real-world situations, such as the relationship between the distance traveled and the time taken, or the relationship between the cost of an item and the quantity purchased.
The constant of proportionality, $k$, represents the rate of change between the two variables and can be determined by the ratio of the two variables.
Directly proportional relationships are a special case of linear equations, where the $y$-intercept is zero.
Review Questions
Explain how the graph of a directly proportional relationship differs from the graph of an inverse proportional relationship.
The graph of a directly proportional relationship is a straight line passing through the origin, with the two variables moving in the same direction. In contrast, the graph of an inverse proportional relationship is a hyperbola, with the two variables moving in opposite directions, and their product remaining constant. While a directly proportional relationship has a constant ratio between the variables, an inverse proportional relationship has a constant product between the variables.
Describe the relationship between the constant of proportionality and the slope of the line in a directly proportional relationship.
In a directly proportional relationship, the constant of proportionality, $k$, is equal to the slope of the line. The slope represents the rate of change between the two variables, and it is constant throughout the entire relationship. The equation $y = kx$ shows that the constant of proportionality, $k$, is the value that the independent variable, $x$, is multiplied by to obtain the dependent variable, $y$. This constant rate of change is reflected in the slope of the line, which is also equal to $k$.
Explain how a directly proportional relationship can be used to model real-world situations, and provide an example.
Directly proportional relationships are commonly used to model real-world situations where two variables are related by a constant rate of change. For example, the relationship between the distance traveled and the time taken for a constant speed can be modeled using a directly proportional relationship. If a car travels a distance of $d$ kilometers in $t$ hours, the relationship can be expressed as $d = kt$, where $k$ is the constant of proportionality representing the car's speed. This model can be used to predict the distance traveled for a given time or the time required to travel a certain distance, as long as the speed remains constant.
An inverse proportion is a relationship between two variables where as one variable increases, the other variable decreases at a constant rate. The variables move in opposite directions, and the product of the two variables remains constant.
The constant of proportionality, also known as the proportionality constant, is the value that the two directly proportional variables are multiplied by to maintain the constant ratio between them.
A linear equation is an equation that represents a directly proportional relationship, where the variables are related by a constant rate of change, forming a straight line when graphed.