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

You are watching: What is the difference between slope and rate of change

This can be used to numerous real life situations.

Take a look in ~ the following graph.


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

John might want to analyze his finances a little more and figure out around how much he was conserving per month. This is dubbed the price of change per month.

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

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

We will must use a formula for finding slope offered two points.


Slope Formula


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


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Let"s take it a look in ~ John"s graph again. John would prefer to find out exactly how much money he conserved per month because that the year.

In other words, John desires to understand the rate of change every month. We room finding out exactly how much John"s account alters per month (on average).

We check out that his beginning balance is $300. ~ above the graph, this allude is (0,300)

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

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

We have the right to now use the steep formula to uncover the steep of the line. The steep is the rate of readjust from one month come the next.

Take a look at at just how this deserve to be solved.


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The steep is equal to 100. This means that the price of change is $100 every month.

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

This gives us one "overview" the John"s savings every month.

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

Example 2: price of Change

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

For this problem, us don"t have a graph to describe in order to identify the 2 ordered pairs. Therefore, us must discover two ordered pairs within the paper definition of this problem.

I am provided information about the year in which Linda to buy a house and also the amount the the home is worth. Due to the fact that these two items room related, I have the right to write them together an notified pair.


Special Note:

If time is connected (time the day, months, years...) the will always be her x coordinate!

Time is always an x value.


Another thing that i would choose to suggest out is the explain (Let x = 0 stand for 1990)

*Believe the or not, mathematicians don"t like to job-related with large numbers. So, instead of working through the yes, really year, we space going to use a substitution. The says, allow x = 0 represent 1990. This is most most likely the early stage year or the year the home was built.

The substitutions are as follows:

0 = 1990

1 = 1991

2 = 1992

3 = 1992

And therefore on...

Let"s solve.

Solution

Let x = year

Let y = amount

Step 1: Write two ordered pairs:

(8, 144,000)      (In 1998, she purchase 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.


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Linda"s average annual rate of readjust if $9,182 dollars per year.

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

Now let"s take a look in ~ one much more example wherein all us are given is a graph. We should pay close fist to the graph in bespeak to deal with the problem.

Let"s take it a look.

Example 3: examining a Graph to identify Rate of Change


The following graph represents Karen"s Marathon.

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

2. Define what you think may have happened throughout interval C.

3. If the rate of change for expression A had remained consistent throughout the whole marathon, exactly how long would it have taken Karen to end up the marathon? (There are 26 mile in a marathon).


Solution

1. What is the price of change for term A?

Notice the interval is from the beginning to 1 hour.

Step 1: identify the 2 points the cover interval A.

The very first point is (0,0) and the 2nd point is (1,6).

Step 2: use the slope formula to uncover the slope, i beg your pardon is the rate of change.


2. Describe what girlfriend think may have happened during interval C.

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

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


The three examples over demonstrated three different ways the a price of adjust problem may be presented.

Just remember, that price of readjust is a method of asking for the steep in a real world problem. Real life troubles are a little an ext challenging, yet hopefully friend now have actually a far better understanding.

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