Slope and Rate of Change (2024)

Why do we need to find the slope of a line in real life?

The slope of a line tells us how something changes over time. If we find the slope we can find the rate of change over that period.

This can be applied to many real life situations.

Take a look at the following graph.

If you've never used this formula before, please visit our page on using the slope formula.

The slope is equal to 100. This means that the rate of change is $100 per month.

Therefore, John saves on average, $100 per month for the year.

This gives us an "overview" of John's savings per month.

Let's take a look at another example that does not involve a graph.

Example 2: Rate of Change

In 1998, Linda purchased a house for $144,000. In 2009, the house was worth $245,000. Find the average annual rate of change in dollars per year in the value of the house. Round your answer to the nearest dollar. (Let x = 0 represent 1990)

For this problem, we don't have a graph to refer to in order to identify the two ordered pairs. Therefore, we must find two ordered pairs within the context of this problem.

I am given information about the year in which Linda purchased a house and the amount that the house is worth. Since these two items are related, I can write them as an ordered pair.

Another thing that I would like to point out is the statement (Let x = 0 represent 1990)

*Believe it or not, mathematicians don't like to work with large numbers. So, instead of working with the actual year, we are going to use a substitution. It says, let x = 0 represent 1990. This is most likely the initial year or the year the house was built.

The substitutions are as follows:

0 = 1990

1 = 1991

2 = 1992

3 = 1992

And so on...

Let's solve.

Solution

Let x = year

Let y = amount

Step 1: Write two ordered pairs:

(8, 144,000) (In 1998, she purchased the house for $144,000)

(19, 245,000) (In 2009 (19 years after 1990) the house is worth $245,000)

Step 2: Use the slope formula to find the slope.

Linda's average annual rate of change if $9,182 dollars per year.

This means that on average, the value of her house increased by $9,182 dollars per year.

Now let's take a look at one more example where all we are given is a graph. We must pay close attention to the graph in order to solve the problem.

Let's take a look.

Example 3: Analyzing a Graph to Determine Rate of Change

Solution

1. What is the rate of change for interval A?

Notice that interval is from the beginning to 1 hour.

Step 1: Identify the two points that cover interval A.

The first point is (0,0) and the second point is (1,6).

Step 2: Use the slope formula to find the slope, which is the rate of change.

2. Explain what you think may have happened during interval C.

During interval C, Karen took a break and stopped running. During that 1/2 hour time period, her distance did not increase.

3. If the rate of change for interval A had remained constant throughoutthe whole marathon, how long would it have taken Karen to finish themarathon? (There are 26 miles in a marathon)

The three examples above demonstrated three different ways that a rate of change problem may be presented.

Just remember, that rate of change is a way of asking for the slope in a real world problem. Real life problems are a little more challenging, but hopefully you now have a better understanding.

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