## Introduction

Understanding proportional relationships through graphing is important in many fields such as finance, economics, and science. Proportional relationships can be represented in different ways, which may cause confusion for those who are just starting to learn about it. In this article, we will explore which graph represents a proportional relationship, by using different examples, both real-life and historical, and provide a step-by-step guide on how to solve practical problems that involve proportional relationships.

## Definition of Proportional Relationships

A proportional relationship is a relationship between two variables that has a constant ratio between them. In terms of math and graphing, this means that for every increase (or decrease) in one variable, the other variable increases (or decreases) by the same proportion. A basic proportional relationship graph is a straight line that passes through the origin, as the ratio between the two variables is constant.

## Using Visual Examples

Graphs that represent proportional relationships are essential tools in analyzing data. For instance, a graph that indicates temperature vs. time would show temperature increasing (or decreasing) as time advances by a specific value. In a proportional relationship, the rates of change must be consistent, indicating that the relationship is proportional. A prime example of a proportional relationship graph is the length vs. the perimeter of a square. As the dimensions of a square increase proportionally, so does its perimeter.

## Comparison of Proportional and Non-Proportional Relationships

It is essential to differentiate proportional and non-proportional relationships. Non-proportional relationships are those that do not have a consistent rate of change between the two variables. One prime example of a non-proportional relationship is the distance vs. time for a car accelerating at a varying speed. The graph will not be a straight line, as the rate of change between time and distance is not consistent. An upward-curved line indicates acceleration in the car’s speed,

which is not proportional to distance traveled.

## Historical Examples

The history of proportional relationships dates back to the ancient Greeks, who first explored the concept of proportionality in geometry. A historical example that demonstrates proportional relationships is Da Vinci’s famous drawing, the Vitruvian Man. The drawing shows the proportional relationship between human body parts, such as the arm span and the height of an individual. The relationship between different lengths of the human body can be represented in a graph, demonstrating that ratios between body parts are proportional.

## Real-Life Examples

Proportional relationships can be seen in many everyday situations, such as calculating the price of goods over time and analyzing cost and profit in a business venture. One real-life example of a proportional relationship is calculating the price of gasoline and the number of gallons purchased. When the price of gasoline increases proportionally, more money is spent on the same amount of gasoline, so the graph demonstrating gas prices vs. gallons shows a straight line passing through the origin, indicating that the relationship is proportional.

## Solving Practical Problems

One of the essential applications of proportional relationships is in solving practical problems. To solve practical problems involving proportional relationship, the first step is to identify the constants. For instance, if the problem involves the weight of an object and its weight per unit, then the unit weight(linear metre, kilometers or grams) is the constant that needs to be recognized. Then, create a table using the information from the problem, which helps to determine how weight changes with increasing units. Finally, graph the relationship and pay attention to the slope of the resulting line to determine if the relationship is proportional.

## Conclusion

Proportional relationships play an essential role in various fields, including finance, economics, and science. Understanding which graph represents a proportional relationship is crucial to analyzing data effectively. We have explored different examples of proportional relationships, provided examples of graphs that represent proportional and non-proportional relationships, and demonstrated how to solve practical problems involving proportional relationships. Therefore, when you are faced with proportional relationship problems, you can solve them confidently and efficiently.

Remember always to identify the constants, create a table using the data provided, graph the relationship, and determine if it is a proportional relationship.

Don’t hesitate to practice graphing proportional relationships yourself. It is the best way to master them and be confident in using them in practical situations.