How to Use Graphs to Write Proportional Relationship

Step 1: Identify the Variables

Step 2: Graph the Data

Step 3: Draw a Line of Best Fit

A graph shows a proportional relationship when it is a straight line through the origin (0, 0). Its equation is y = kx, where k is the constant of proportionality: pick any point (x, y) on the line other than the origin and compute k = y/x. For example, if the line passes through (4, 10), then k = 10 ÷ 4 = 2.5 and the equation is y = 2.5x.

Next, draw a line of best fit through your data points. For a perfect proportional relationship, all points should lie on this line. However, in the real world, it’s often the case that data are merely “approximately” proportional, so don’t be surprised if your data don’t fit perfectly.

Step 4: Identify the Origin Point

Check whether the line passes through the origin \((0,0)\). If it does, this means that when you have zero of one variable (no apples), you have zero of the other variable (no cost), confirming a proportional relationship.

Step 5: Calculate the Slope

The slope of the line represents the rate of change or the ratio between the two variables. It can be calculated by choosing two points on the line and using the formula \(\frac{(change\:in\:y)}{(change\:in\:x)}\). In our example, the slope would be the cost per apple.

Step 6: Write the Equation

Once you have the slope, you can write the equation of the line. A proportional relationship can be written in the form \(y = mx\), where \(m\) is the slope. In our example, if the slope was \(2\) (meaning each apple costs \($2)\), our equation would be \(y = 2x\).

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Step 7: Interpret the Equation

Finally, interpret the equation in the context of the problem. In our example, \(y = 2x\) means that the total cost \((y)\) is \($2\) times the number of apples \((x)\).

Remember that a proportional relationship means that the ratio between the two variables is always the same. So in our example, no matter how many apples we buy, the cost per apple remains constant at \($2\). This relationship would be represented on the graph as a straight line passing through the origin with a slope of \(2\).

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Related: How to Create a Proportion