Rate of Change

Rate of change is an important mathematical concept. But what exactly is the rate of change? How can we find it? What can it teach us about math? Let's find out:

Rate of change, explained

A rate of change is simply how fast something is changing in relation to something else. We can express this ratio as:

Rate of change = change in y/change in x

For example, what happens if we plant one apple seed? Eventually, we would get an apple tree. This apple tree would produce even more apples as a result. Let's assume that we figured out that one apple seed leads to 200 apples within 5 years. We now know our rate of change.

The above is an example of a positive rate of change. Planting more seeds leads to more apples. But what happens if we cut down a tree instead? This would lead to a net loss of apples in the future. In other words, it is an example of a negative rate of change.

What happens if we neither plant new trees nor cut them down? The number of produced apples would stay the same each year. In other words, this is an example of zero rate of change.

Visualizing rate of change

It's easier to understand how rate of change works if we visualize it. Here's what positive rate of change looks like when we graph it:

As we can see, x and y have a positive relationship. When the value of x increases, so too does the value of y. The same can be said in reverse.

What about negative rate of change? Take a look:

As we can see, the value of y decreases when the value of x increases. This is why we see a characteristic downward slope.

Finally, we have zero rate of change:

Nothing is changing in this example, so the values of x and y stay exactly the same. This is why we see a characteristic straight line.

A real-world example of rate of change

One of the most ubiquitous examples of rate of change is the relationship between time and distance. The longer we travel, the further we get. If we're traveling at a constant speed, we can use this rate of change to determine how far we will travel.

Let's use our knowledge to solve a real-world problem. We can assume the following:

• We can drive 80 miles in 2 hours
• We can drive 160 miles in 4 hours
• We can drive 240 miles in 6 hours

Let's start by establishing our rate of change. We know that the formula for rate of change is the following:

Rate of change is defined as the change in y divided by the change in x.

$\frac{160-80}{4-2}=\frac{80}{2}=\frac{40}{1}$

Now we know that our rate of change is $\frac{40}{1}$ or $40$ . This tells us that our car is traveling at 40 miles per hour. We can graph this rate of change by plugging in our known values and connecting the dots:

Using the rate of change formula

You can use the rate of change formula to make many important calculations.

• The increase of current in amperes is based on the increase in voltage
• Spotting security momentum and other trends in the financial world
• How much work you can get done within a specific amount of time
• How many people are required to get a specific job done in a certain time
• Projecting future earnings or saving your money

With all that said, you will notice that many equations related to rates of change involve time. This is because time is one of the most important constants in our universe.

Topics related to the Rate of Change

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