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Course: Algebra 2 > Unit 1
Lesson 2: Average rate of change of polynomialsFinding 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?(62 votes)
[-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.(56 votes)
i m genuinely confused, i don't understand the topic.(23 votes)
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.(32 votes)
Can somebody explain the graph that he drew? I understood everything up to that point.(12 votes)
The graph Sal drew was supposed to help visualize the average rate of change (aka the slope).(3 votes)
At around1:30, Sal starts talking about the chart, but when he was multiplying, I did not understand how he got y=f(x)=-8+8=0(9 votes)
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.(13 votes)
How did I go from definitions of words used in polynomials to THIS. I don't understand anything -_-.(10 votes)- For me it's also helpful to mentally plug in f(x)=y so I understand where the x and y comes from(1 vote)
At0:14, 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”?(4 votes)
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).(6 votes)
- So on the graph, I get the change in y and the change in x. What I don't get is the loopy line, and what defines the loopy line. The most I can make of it is the start and end. But couldn't we just use a straight line? Why does it have to be so curvy, and what purpose does it serve?(3 votes)
Well for this example, that loopy line is a polynomial function. More specfically, it's f(x) = x³-4x. Any type of polynomial with an degree (highest exponent) as two or more are naturally curvy. Desmos would help you lots + plugging numbers in !
The purpose of curvy lines and not straight lines, is not everything is going to be as linear as we may thing, world speaking. Mathematics does not only cover just straight lines, but curvy, fulling,weirdly interesting functions very philosophical lmao but hopefully that helps.(7 votes)
- I don't really understand.. what is "average rate of change.."?(4 votes)
- The "average rate of change" is a concept in calculus that measures how a quantity changes on average over a certain period of time. It's calculated by dividing the change in the quantity by the change in time.(4 votes)
why does sal make y = f(3) = 27-12=15. Where does the 15 come from. The other part I understand.(3 votes)- The original equation is f(x) = x³-4x.
f(3) is simply replacing everything you see with x with 3; thus you'd get:
f(3) = 3³-4(3)
= 27 - 12
= 15
f(3) = 15, hopefully that helps!(6 votes)
- Ok hear me out. The equation of rate of change is
f(b)-f(a)/b-a (this wasn't shown in the video)
you can also think of this as Δy/Δx
According to this equation we would get 15-0/3-2 which then simplifies into 15/1 = 15. This is because Δx = xf-xi which is 3-2 in the denominator.
But in this explanation (At2:05) he uses the amount of numbers in between -2 and 3 as if you where counting on a numberline. from that he gets 15/5 =3 and not 15/1.
But how is that method right? You are supposed to look for the difference between X2 and X1 not the amount of numbers between them. Δx literally means Xf-xi or x2-x1 not what is shown here.
So can someone tell me why he is solving it like this? Have i been lied to all my life and this is somehow correct? or is the rate of change equation inapplicable in here and this method here is right?(2 votes)
His method is correct. It can be confusing as he arithmetically obtained the answer. Also,
xf - xi = 3 - (-2) = 3 + 2 = 5
when subtracting integers, you must use the CCC method, in which you change the operation and sign of the second number. Hope this helps.(6 votes)
1:30Video transcript
- [Instructor] We are asked, What is the average rate of
change of the function f, and this function is f up here, this is the definition
of it, over the interval from negative two to three,
and it's a closed interval because they put these brackets around it instead of parentheses, so that means it includes
both of these boundaries. Pause this video and see if
you can work through that. All right, now let's
work through it together. So there's a couple of ways
that we can conceptualize average rate of change of a function. One way to think about it is it's our change in the
value of our function divided by our change in x, or it's our change in
the value of our function per x on average. So you can view it as change
in the value of the function, divided by your change in x. If you say that y is equal to f of x, you could also express it as change in y over change in x. On average, how much
does the function change per unit change in x on average? And we could do this with a table, or we could try to
conceptualize it visually. Let's just do this one with a table, and then we'll try to
connect the dots a little bit with a visual. So if we have x here, and
then if we have y is equal to f of x right over here, when
x is equal to negative two, what is y going to be equal to, or what is f of negative two? Well, let's see. f of, so why is equal
to f of negative two, which is going to be
equal to negative eight, that's negative two to the third power, minus four times negative two, so that's minus negative
eight, so that's plus eight. That equals zero. And then when x is equal to three, I'm going to the right
end of that interval. Well, now y is equal to f of three, which is equal to 27,
three to the third power, minus four times three, minus
12, which is equal to 15. So what is our change in y over our change in x over this interval? Well, our y went from zero to 15, so we have an increase of 15 in y. And what was our change in x? Well, we went from negative
two to positive three, so we had a plus five change in x. So our change in x is plus five, and so our average rate of change with y with respect to x, or the
rate of change of our function with respect to x, over the interval, is going to be equal to three. If you wanted to think
about this visually, I could try to sketch this. So this is the x-axis, the y-axis, and our function does something like this. So at x equals negative
two, f of x is zero, and then it goes up, and
then it comes back down, and then it does something like this, it does something like this, and it does, and it was going before this, and so the interval that we care about, where we're going from
negative two to three, which is right about there. So that's x equals negative
two to x equals three, and so what we want to do, at
the left end of the interval, our function is equal to
zero, so we're at this point, right over there, I'll
do this in a new color, at this point, right over there, and at the right end of our function, f of three is 15, so we
are up here someplace. Let me connect the curve a little bit. We are going to be up there. And so when we're thinking
about the average rate of change over this interval, we're
really thinking about the slope of the line that
connects these two points. So the line that connects these two points looks something like this. And we're just calculating
what is our change in y, which is going to be
this, our change in y, and we see that the value of our function increased by 15, divided
by our change in x. So this right over here
is our change in x, which we see went from
negative two to three. That's going to be equal to five. So that's all we're doing when we're thinking about
the average rate of change.