Directly proportional graph

Here you will learn about directly proportional graphs as well as inversely proportional graphs, including what proportion graphs are and how to interpret them.

Students will first learn about directly proportional graphs as part of functions in high school.

What are directly proportional graphs?

Directly proportional graphs and inversely proportional graphs represent the two types of proportional relationships commonly studied in high school: direct proportion and inverse proportion.

To determine the graph of a directly proportional relationship or an inversely proportional relationship, you need to understand how the two variables are linked and be able to sketch this on a set of axes.

Directly proportional graphs

Direct proportion is one type of proportionality relationship. As one value increases, so does the other value. Direct proportion is sometimes known as direct variation.

Direct proportion is useful in numerous real life situations such as exchange rates, buying a quantity of items, work rates, and conversion between units.

For example, the earnings received from selling a greater amount of the same product increases in direct proportion to the number of products sold.

The symbol ∝ \propto ∝ represents a proportional relationship.

If y is directly proportional to x, you write it as y∝x.

There is also a direct proportion formula that uses the constant k as the constant of proportionality:

Step-by-step guide: Direct variation formula

Directly proportional graphs can be straight line graphs:

As the variable x increases, the y variable increases, so the straight line continues on the same gradient as x gets larger; conversely, the straight line tends to 0 as x and y get smaller.

The straight line has a positive gradient, k where k=y/x, and the line intersects the origin (0,0).

Inversely proportional graphs

Inverse proportion is when as one quantity increases the other quantity decreases and vice versa. Inverse proportion is sometimes known as indirect proportion.

For example, consider a job such as painting a fence. If the number of workers increases, then the time taken to do the work decreases. If the number of workers decreases, then the time taken to do the work increases.

If y is inversely proportional to x then you write y∝1/x.

There is also an inverse proportion formula that uses the constant k as the constant of proportionality. The inverse proportion formula is:

y= k/x.

Inversely proportional graphs are reciprocal graphs consisting of one smooth curve in the first quadrant only.

As the variable x increases, the y variable decreases and so the curve tends towards the x-axis as x gets larger; conversely, the curve tends towards the y-axis as x gets smaller. The curve does not touch or intersect either axis at any point.

Common Core State Standards

How does this relate to high school math?

  • High School – Functions – Interpreting Functions (HS.F.IF.C.7) Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases.

How to recognize a directly proportional graph

In order to recognize a directly proportional graph or inversely proportional graph:

  1. Determine the type of proportional relationship.
  2. Check the key features of the proportion graph.

Teaching tips for directly proportional graphs

  • Use worksheets with relatable examples of direct proportion, such as:
    • Speed and distance (constant speed: distance = speed × time).
    • Price and quantity (buying more items increases cost proportionally).
  • Teach students that in a directly proportional graph, the values on the x-axis (independent variable) and the y-axis (dependent variable) always maintain a consistent ratio.
  • Emphasize that the graph is a straight line that starts at the origin (0,0), showing that when the value of x is zero, y is also zero. This visual connection helps reinforce the concept of proportionality.