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## Integrated math 3

### Course: Integrated math 3>Unit 1

Lesson 2: Average rate of change of polynomials

# Finding average rate of change of polynomials

Learn how to calculate the average rate of change of a function over a specific interval. Discover how changes in the function's value relate to changes in x. Use tables and visuals to understand the concept better. This is key to mastering polynomial functions in algebra.

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• Okay, this is over my head. In my school we take Algebra 1, then Geometry, then Algebra 2 (This upcoming year I’m going to take Algebra 2). So that may be why I’m so confused. How does [-2,3] affect what he did to solve the problem? And I don’t really get why he used the table to get the answer? And what he was looking for in the table? Can somebody explain or know a video that could help me understand? •  [-2,3] is fancy math notation. The [ ] means you include those values, so the interval we want to examine goes from -2 to +3. It really means: plug -2 in for x and find the y value. Then plug 3 in for x and find the y value. How much does the input (x-value) change from -2 to 3? (it changes by +5). How much does the output (y value) change over the x interval of -2 to 3? (it changes by +15). Average rate of change means change in output (change in y, or f(x)), divided by change in input (change in x). Sal used a table to show this algebraically. Using algebra means you don't have to know what the function looks like on a graph; you can still do the problem, BUT being able to see it with algebra and visually (on a graph) will allow for a deeper understanding and more connection of what the numbers mean.
• i m genuinely confused, i don't understand the topic. • Hello! I hope this helps.

The definition of a Function is "a relationship or expression involving one or more variables" (credits to the Oxford English dictionary and Google). Non-linear functions are more confusing than linear functions because they are less concrete and, well, linear than an Algebra I function.

A Function is the relation of things through input and output. The most common name for a function is (f), but it can be anything. What goes into a function (the input) is put in parentheses, such as f(x). The X is just a placeholder for a number. If you get the function f(x) = x^2, you plug in the numbers that they give you. Here is an example of the Function listed above if you plug in the number 5:
f(5) = 5^2.

You can also make a function look like this:
y=x^2
There is no name for the Function, but there is an input (x), a relationship between the two numbers (^2), and an output (y). The equation above is a function because a function relates an input to an output.

To test if a graphed line or curve is a function, make sure that the line has no vertical parts, and having part of it go through the same x point. For example, having something go through the point (7,8), and then going through the point (7,9). A Function having two inputs is a valid curve or line, but not a function. HOWEVER, you can have two repeated y points. For example, having one going through the points (9,8) and then having them go through the points (7,8). That means that there can be multiple outputs, but only one input.

Functions are polynomials, given that they relate to one or more variables (this allows for monomials - things with only one number, like 8). Sal is talking about how to find the average rate of change (like a slope) for a non-linear function.

Hope this helps, Mimi (and anyone else with that question). Let me know if anything was unclear.
• Can somebody explain the graph that he drew? I understood everything up to that point. • i have no idea what i just watched .... • At around , Sal starts talking about the chart, but when he was multiplying, I did not understand how he got y=f(x)=-8+8=0 • f(x) = x^3 - 4x

so in other words you plug in numbers for x to find what y is since y = f(x). In this instance Sal wanted t find f(-2). Or in other words, if you graphed f(x) where is the line of the graph when x=-2

All this means is that you take f(x) and plug -2 in for all of the xs. the way this is written out is f(-2)

so f(x) = x^3 - 4x so if you plug in -2 for all xs this gets us (-2)^3 - 4x. (-2)^3 = -2*-2*-2 = 8 and -4*-2 = 8 so (-2)^3 - 4x = -8 + 8 = 0.

Let me know if that didn't help.
• How did I go from definitions of words used in polynomials to THIS. I don't understand anything -_-. • I don't really understand.. what is "average rate of change.."? • At , what is the difference between an interval with brackets to an interval without brackets? What would an interval without brackets look like and how would it change how the function is “solved”? • There are different types of brackets. They can be open or closed.
In the video, the brackets used were '[]'. These are considered closed brackets and are inclusive to the numbers in them. For example, with [-2, 3] it's asking about the range of -2 to 3, including -2 and 3.
There can also be open brackets - '()'. These are exclusive and don't include the numbers in them. If it were (-2, 3), it'd be asking about the range between -2 and 3, not including -2 or 3.
You can also have things such as (-2, 3] (excluding -2 but including 3) or [-2, 3) (including -2 but excluding 3).
• why does sal make y = f(3) = 27-12=15. Where does the 15 come from. The other part I understand.  