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

You are watching: Is there a connection between rate and slope

This can be used to plenty of real life situations.

Take a look at the adhering to graph.


This graph shows exactly how John"s save account balance has readjusted over the course of a year. We have the right to see the he opened up his account v $300 and by the finish of the very first monthhe had actually saved $100. By the end of the 12 month time span, John had $1500 in his savings account.

John might want to analysis his finances a little more and number out around how lot he was conserving per month. This is referred to as the rate of change per month.

By finding the slope of the line, we would certainly be calculating the price of change.

We can"t count the rise over the run favor we walk in the calculating steep lesson since our devices onthe x and y axis room not the same. In most real life problems, your devices will no be the same on the x and also y axis. So, us need another method!

We will should use a formula for finding slope provided two points.

Slope Formula


If you"ve never used this formula before, you re welcome visit our page on making use of the slope formula.


Let"s take it a look at John"s graph again. Man would favor to find out exactly how much money he saved per month because that the year.

In various other words, John wants to know the price of change every month. We are finding out just how much John"s account changes per month (on average).

We see that his beginning balance is $300. On the graph, this suggest is (0,300)

His finishing account balance (on month 12) is $1500. This suggest is (12, 1500).

Therefore, our two ordered pairs room (0,300) and (12, 1500).

We deserve to now use the slope formula to discover the slope of the line. The slope is the rate of adjust from one month to the next.

Take a watch at how this can be solved.


The steep is equal to 100. This method that the rate of readjust is $100 per month.

Therefore, John conserves on average, $100 every month because that the year.

This offers us one "overview" of John"s savings per month.

Let"s take a watch at one more example that does not involve a graph.

Example 2: rate of Change

In 1998, Linda purchased a home for $144,000. In 2009, the residence was worth $245,000. Find the average annual rate of readjust in dollars per year in the worth of the house. Round her answer come the nearest dollar. (Let x = 0 stand for 1990)

For this problem, we don"t have actually a graph to describe in bespeak to identify the two ordered pairs. Therefore, we must discover two ordered bag within the paper definition of this problem.

I am provided information around the year in i m sorry Linda purchased a house and the amount the the residence is worth. Because these two items room related, I can write them as an ordered pair.

Special Note:

If time is affiliated (time of day, months, years...) the will constantly be your x coordinate!

Time is always an x value.

Another point that ns would prefer to suggest out is the statement (Let x = 0 represent 1990)

*Believe that or not, mathematicians don"t like to work-related with big numbers. So, instead of working v the yes, really year, we are going to use a substitution. It says, permit x = 0 represent 1990. This is most likely the early stage year or the year the home was built.

The substitutions space as follows:

0 = 1990

1 = 1991

2 = 1992

3 = 1992

And therefore on...

Let"s solve.


Let x = year

Let y = amount

Step 1: Write 2 ordered pairs:

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

(19, 245,000)    (In 2009 (19 year after 1990) the home is precious $245,000)

Step 2: use the steep formula to find the slope.


Linda"s average yearly rate of readjust if $9,182 dollars every year.

This means that top top average, the worth of her home increased by $9,182 dollars per year.

Now let"s take it a look at one much more example where all us are given is a graph. We should pay close attention to the graph in order to resolve the problem.

Let"s take a look.

Example 3: evaluating a Graph to recognize Rate that Change

The following graph to represent Karen"s Marathon.

1. What is the price of readjust for expression A?

2. Explain what friend think may have actually happened throughout interval C.

3. If the rate of adjust for term A had remained continuous throughout the totality marathon, just how long would it have taken Karen to finish the marathon? (There are 26 mile in a marathon).


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

Notice the interval is from the start to 1 hour.

Step 1: recognize the two points the cover interval A.

The an initial point is (0,0) and also the second point is (1,6).

Step 2: use the steep formula to find the slope, i m sorry is the rate of change.

2. Explain what friend think may have actually happened throughout interval C.

During expression C, Karen take it a break and stopped running. During that 1/2 hour time period, her street did no increase.

3. If the rate of readjust for expression A had remained constant throughout the whole marathon, exactly how long would certainly it have actually taken Karen to complete the marathon? (There room 26 miles in a marathon)

The 3 examples above demonstrated three different ways the a price of change problem may be presented.

Just remember, that rate of adjust is a way of questioning for the steep in a real people problem. Actual life problems are a little an ext challenging, yet hopefully girlfriend now have actually a far better understanding.

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