Interpret Rate of Change and Initial Value

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  • 0:02 Rate of Change & Initial Value
  • 1:41 Interpreting Rate of Change
  • 2:30 Word Problems
  • 4:02 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

In this lesson, you'll learn how to interpret the rate of change and initial value of a function. We'll also discuss applying these principles to word problems and in real-world situations.

Rate of Change & Initial Value

The initial value of a function is the output value of the function when the input value is 0. For example, if your function is tracking how much money you made over a certain amount of time, then the initial value would be however much money you had to begin with on day 0.

The rate of change of a function is how fast the output changes relative to the input. On a graph, this is the same thing as how fast the y-value changes relative to the x-value. Here's an example so you can see how this works.

Gilly the Gerbil Gal runs an at-home grooming service for pampered gerbils. This is the function that Gilly uses to determine the cost of her gerbil grooming service, where y represents the cost of grooming, and x represents the number of gerbils being groomed: y = 10 + 5x.

Table of Prices

The table above shows the prices for hiring Gilly the Gerbil Gal to groom your pet gerbils. If you just look at this table of values, you can see that the initial value is $10 - that's the base cost for Gilly to come out to your house. Gilly then charges $5 for each gerbil groomed - that's the rate of change: $5 per gerbil.

A graph of the function values is based on the inputs and outputs
Graph of function

Here's what that would look like on a graph. You can see that initial y-value is $10 - that's the initial value of the function, or the value when x = 0. For every 1 unit of change on the x-axis there is a $5 change on the y-axis, so the rate of change is $5 per gerbil.

Interpreting Rate of Change

Once you know what the rate of change is, you can use it to determine other information about the function. One thing you can do is to use the rate of change to make predictions about the function's behavior for other input values. For example, you could easily find out what Gilly would charge for grooming 6 gerbils. Since the rate of change is $5 for every gerbil, just add $5 to the cost for 5 gerbils, and you'll end up with $40 for 6 gerbils.

You can also use the rate of change to find the initial value if you don't know it. For example, if you just got this, you would be able to tell the rate of change is $5 per gerbil, and so you could work backwards to find the initial value, which is the price for just a house visit, or $10.

Word Problems

Now it's time to talk about how to tackle problems about rate of change and initial value if they're presented as word problems. Hiding the math inside a word problem can make it tricky to figure out what's going on, but one simple solution is to make either a table or a graph of the values in the word problem. Then, instead of trying to intuit which numbers are what, you can simply look at the function in an easier form. Here's an example:

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