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Properties of Functions
- Over an interval of x-values, a given function can be increasing, decreasing, or constant. That is, always going up, always going down, or not changing, respectively. This idea is easiest to understand visually, so look at a graph to find where these things occur.
- We talk about increasing, decreasing, and constant in terms of intervals: that is, sections of the horizontal axis. Whenever you talk about one of the above as an interval, you always give it in parentheses.
- An (absolute/global) maximum is where a function achieves its highest value. An (absolute/global) minimum is where a function achieves its lowest value.
- A relative maximum (or local maximum) is where a function achieves its highest value in some "neighborhood". A relative minimum (or local minimum) is where a function achieves its lowest value in some "neighborhood". [Notice that these aren't necessarily the highest/lowest locations for the entire function (although they might be), just an extreme location in some interval.]
- We can refer to all the maximums and minimums of a function (both absolute and relative) with the word extrema: the extreme values of a function.
- We can calculate the average rate of change for a function between two locations x_{1} and x_{2} with the formula
f(x_{2}) − f(x_{1}) x_{2} −x_{1}. - It is often very important to know what x values for a function cause it to output 0, that is to say, f(x) = 0. This idea is so important, it goes by many names: the zeros of a function, the roots, the x-intercepts. But these all mean the same thing: all x such that f(x) = 0.
- An even function (totally different from being an even number) is one where
In other words, plugging in the negative or positive version of a number gives the same output. Graphically, this means that even functions are symmetric around the y-axis (mirror left-right).f(−x) = f(x). - An odd function (totally different from being an odd number) is one where
In other words, plugging in the negative version of a number gives the same thing as the positive number did, but the output has an additional negative sign. Graphically, this means that odd functions are symmetric around the origin (mirror left-right and up-down).f(−x) = − f(x).
Properties of Functions
- Increasing, decreasing, and constant behave like they sound. Assuming you are "reading" the graph from left to right, an increasing interval means it is going up, decreasing means it is going down, and constant means the height does not change.
- To find the increasing intervals, we look on the graph to see where the function is going up. To find decreasing, we find where it is going down. To find constant, we find where it does not change.
- Write each of these intervals in interval notation, such as (−5, −3). With increasing/decreasing/constant intervals, we always use parentheses, never brackets, because we can't include where the function "turns" in the interval.
At what x-values do the absolute maximum and minimum occur? What are the values for the maximum and minimum?
- Increasing, decreasing, and constant behave like they sound. Assuming you are "reading" the graph from left to right, an increasing interval means it is going up, decreasing means it is going down, and constant means the height does not change.
- To find the increasing intervals, we look on the graph to see where the function is going up. To find decreasing, we find where it is going down. To find constant, we find where it does not change. Write these intervals in interval notation and only using parentheses ( ). [Notice that this function has no constant intervals.]
- The absolute maximum is the highest value achieved by the function. The absolute minimum is the lowest value achieved by the function.
- To find the max/min, look on the graph for the highest/lowest points. The horizontal location of these points gives the x-value, and the height gives the value for the max/min.
Absolute Maximum: at x=3⇒ f(3) = 4; Absolute Minimum: x=−1 ⇒ f(−1) = −4
At what x-value do we find a relative maximum? At what x-value do we find a relative minimum? What are the values of those maximums and minimums?
Explain why the function does NOT have an absolute maximum or minimum.
- Increasing, decreasing, and constant behave like they sound. Assuming you are "reading" the graph from left to right, an increasing interval means it is going up, decreasing means it is going down, and constant means the height does not change.
- The graph helps us find the increasing/decreasing/constant intervals, but it doesn't tell the whole story. Remember, we were told that it is a graph of f(x) = x^{3} + 3x^{2} −1. Unlike previous problems, f(x) is not confined to this graphing window. It keeps going beyond the edges of this graph.
- The graph will go down forever as it goes off to the left, while it will go up forever as it goes to the right. This means we will need to use ∞ in our interval notations.
- The relative maximum is the highest value achieved by the function in some neighborhood. The relative minimum is the lowest value achieved by the function in some neighborhood. Relative max/min don't have to to be the highest or lowest for the whole function, just the highest or lowest in some local "zone" of the function.
- Looking at the graph, we can see an upward bump and a downward bump. The points at the tips of these bumps give us the relative maximum and minimum. The horizontal location of these points gives the x-value, and the height gives the value for the max/min.
- To have an absolute max/min, the function must achieve a value that is higher/lower than all the other values. On this function though, can that ever happen? If you were to name a highest value, would be able to find another value that surpasses it?
Relative Maximum: at x=−2⇒ f(−2) = 3; Relative Minimum: x=0 ⇒ f(0) = −1;
The function cannot have an absolute max/min because it continues to travel up and down forever. There is no single highest/lowest location, because f(x) will always go above/below any value.
- The roots of a function (also called the zeros or x-intercepts) are all the x-values such that f(x) = 0. In other words, they are the locations that cause the function to output 0.
- Graphically, we can see this as where the function crosses the x-axis, because that height is equivalent to f(x)=0.
- The zeros of a function (also called the roots or x-intercepts) are all the x-values such that g(x) = 0. In other words, they are the locations that cause the function to output 0.
- Previously, we found the zeros of a function by looking at its graph and seeing where it crossed the x axis. Potentially, we could do that here by carefully graphing g(x), but that would be difficult and time-consuming. Instead, we can do it algebraically.
- Since a zero (root) is an x-value where g(x) = 0, we just set g(x) to 0 in the function, then solve the resulting equation for x:
0 =
√2x+5−5
- The average rate of change is the slope of an imaginary line between two points on the graph of the function (this line is called the secant line). These two points are determined by the two x locations we are finding the average rate of change between.
- As a formula, the average rate of change is
f(x_{2}) − f(x_{1}) x_{2} −x_{1}. - For the first pair of x-values, we plug them into the average rate of change formula to get
From there, simplify to find the average rate of change.f(2) − f(0) 2 −0= (3·2^{2} +2) − (3·0^{2} + 2) 2. - For the second pair, we do the same thing:
From there, simplify to find the average rate of change.f(4) − f(2) 4 −2= (3·4^{2} +2) − (3·2^{2} + 2) 2.
- The secant line is an imaginary line that passes through two points on a function. The slope of the secant line is equal to the average rate of change over that interval of the function.
- Begin by finding the slope of the secant line. This will be the same whether we use the formula for the average rate of change or just find the slope between the two points:
Simplify this to find the slope of the secant line.f(4) − f(1) 4−1= (4^{2}−3)−(1^{2}−3) 3 - Once you know the slope of the secant line is m=5, you can find the rest of the equation for the secant line. A useful and common formula for a line is y=mx +b, where m is the slope and b is the y-intercept. At this point we just need to find b.
- We can plug m=5 into y=mx+b to get y = 5x+b. Now if we know any other point on the line, we can plug it in to the equation, and that will allow us to find b. We know both (1, f(1)) and (4, f(4)) are on the secant line, so let's use one of them. Let's choose (1, f(1)) = (1, −2). Plugging that into our equation, we have
which we can solve for b.−2 = 5 (1) + b,
- An even function is one where f(−x) = f(x). In other words, plugging in the negative version of a number gives the same result as the positive version of a number.
- To show that f(x) is even, we need to show that this property is true for any number. We can do that by considering the arbitrary number x and its negative version −x. Thus, if f(−x) = f(x), we have shown that f(x) is even.
- Set up f(−x) by plugging in −x for x from the original function given:
f(−x) = 3(−x)^{4} − (−x)^{2} - Simplify the above. Remember that if you raise a negative number to an even power, it is turned positive. Thus
which shows that f(x) is even.f(−x) = 3(−x)^{4} − (−x)^{2} = 3x^{4} − x^{2} = f(x),
- An odd function is one where f(−x) = −f(x). In other words, plugging in the negative version of a number gives the opposite result as the positive version of a number.
- To show that f(x) is odd, we need to show that this property is true for any number. We can do that by considering the arbitrary number x and its negative version −x. Thus, if f(−x) = − f(x), we have shown that f(x) is odd.
- Set up f(−x) by plugging in −x for x from the original function given:
f(−x) = (−x) (−x)^{2}+3. - Simplify the above to show that f(−x) = −f(x):
Thus f(x) is odd.f(−x) = (−x) (−x)^{2}+3= −x x^{2}+3= − x x^{2}+3= − f(x).
- A function f(x) is even if f(−x) = f(x). A function f(x) is odd if f(−x) = − f(x).
- To show that f(x) = x^{2} − 4x has neither of these qualities, we must find out what f(−x) is.
- Looking at f(−x) and simplifying, we get
f(−x) = (−x)^{2} − 4(−x) = x^{2} + 4x. - To show that f(x) is not even, compare f(−x) = x^{2} + 4x to f(x) = x^{2}−4x. Since f(−x) ≠ f(x), we have that f(x) is not even.
- To show that f(x) is not odd, compare f(−x) = x^{2} + 4x to − f(x) = −(x^{2}−4x) = −x^{2} + 4x. Since f(−x) ≠ − f(x), we have that f(x) is not odd.
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Answer
Properties of Functions
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Intro
- Introduction
- Increasing Decreasing Constant
- Find Intervals by Looking at the Graph
- Intervals Show x-values; Write in Parentheses
- Maximum and Minimums
- Relative (Local) Max/Min
- Max/Min, More Terms
- Average Rate of Change
- Zeros/Roots/x-intercepts
- Even Functions
- Odd Functions
- Even/Odd Functions and Graphs
- Example 1
- Example 2
- Example 3
- Example 4
- Intro 0:00
- Introduction 0:05
- Increasing Decreasing Constant 0:43
- Looking at a Specific Graph
- Increasing Interval
- Constant Function
- Decreasing Interval
- Find Intervals by Looking at the Graph 5:32
- Intervals Show x-values; Write in Parentheses 6:39
- Maximum and Minimums 8:48
- Relative (Local) Max/Min 10:20
- Formal Definition of Relative Maximum
- Formal Definition of Relative Minimum
- Max/Min, More Terms 14:18
- Definition of Extrema
- Average Rate of Change 16:11
- Drawing a Line for the Average Rate
- Using the Slope of the Secant Line
- Slope in Function Notation
- Zeros/Roots/x-intercepts 19:45
- What Zeros in a Function Mean
- Even Functions 22:30
- Odd Functions 24:36
- Even/Odd Functions and Graphs 26:28
- Example of an Even Function
- Example of an Odd Function
- Example 1 29:35
- Example 2 33:07
- Example 3 40:32
- Example 4 42:34
Precalculus with Limits Online Course
Transcription: Properties of Functions
Hi--welcome back to Educator.com.0000
Today, we are going to talk about properties of functions.0002
Functions are extremely important to math; we keep talking about them, because we are going to use them a lot; they are really, really useful.0005
To help us investigate and describe behaviors of functions, we can talk about properties that a function has.0012
There are a wide variety of various properties that a function can or cannot have.0017
This lesson is going to go over some of the most important ones.0022
While there are many possible properties out there that we won't be talking about in this lesson,0024
this lesson is still going to give us a great start for being able to describe functions,0028
being able to talk about how they behave and how they work.0032
So, this is going to give us the foundation for being able to talk about other functions in a more rigorous way,0035
where we can describe exactly what they are doing and really understand what is going on; great.0039
All right, the first one: increasing/decreasing/constant: over an interval of x-values,0045
a given function can be increasing, decreasing, or constant--that is, always going up, always going down, or not changing.0050
Its number will always be increasing; its number will always be decreasing; or its number will be not changing.0060
And by number, I mean to say the output from the inputs as we move through those x-values.0066
This is really much easier to understand visually, so let's look at it that way.0072
So, let's consider a function whose graph is this one right here: this function is increasing on -3 to -1.0076
We have -3 to -1, because from -3 to -1, it is going up; but it stops right around here.0083
So, it stops increasing on -3; it stops increasing after -1, but from -3 to -1, we see that it is increasing.0091
It is probably increasing before -3, but since all we have been given is this specific viewing window to look through,0099
all we can be guaranteed of is that, from -3 to -1, it is increasing.0104
Then, it is constant on -1 to 1; it doesn't change as we go from -1 to 1--it stays the exact same, so it is constant on -1 to 1.0108
However, it is increasing before -1, and it is decreasing after 1; so it is decreasing on 1 to 3, because we are now going down.0121
So, it continues to go down from 1 on to 3, because we can only be guaranteed up until 3.0131
It might do something right after the edge of the viewing window, so we can only be sure of what is there.0138
It is decreasing from 1 to 3--great; that is what we are seeing visually.0143
It is either going up, straight, or down; it is either horizontal, it is going up, or it is down.0147
Increasing means going up; constant means flat; decreasing means down.0153
Formally, we say a function is increasing on an interval if for any a and b in the interval where a < b, then f(a) < f(b).0160
Now, that seems kind of confusing; so let's see it in a picture version.0167
Let's say we have an interval a to b, and this graph is above it.0172
If we have some interval--it is any interval, so let's just say we have some interval--that is what was between those two bars--0179
and then we decide to grab two random points: we choose here as a and here as b;0188
a is less than b, so that means a is always on the left side; a is on the left, because a is less than b.0195
That is not to say b over c; that is because--I will just rewrite that--we might get that confused in math.0204
a is on the left because a is less than b; if we then look at what they evaluate to, this is the height at f(a), and then this is the height at f(b).0211
And notice: f(b) is above f(a); f(b) is greater than f(a).0224
So, it is saying that any point on the left is going to end up being lower than points on the right, in the interval when it is increasing.0231
In other words, the graph is going up in the interval, as we read from left to right.0240
Remember, we always read graphs from left to right.0245
So, during the interval, we are going from left to right; we are going up as we go from left to right.0248
We have a similar thing for decreasing; we have some interval, some chunk, and we have some decreasing graph.0255
And if we pull up two points, a and b (a has to be on the left of b, because we have a < b), then f(a) > f(b).0265
f(a) > f(b); so decreasing means we are going down--the graph is going down from left to right.0277
We don't want to get too caught up in this formal idea; there is some interval, some place,0287
where if we were to pull out any two points, the one on the left will either be below the one on the right,0293
if it is increasing; or if it is decreasing, it will be above the one on the right.0299
We don't want to get too caught up in this; we want to think more in terms of going up and going down, in terms of reading from left to right.0304
Finally, constant: if we have some interval, then within that interval, our function is nice and flat,0310
because if we choose any a and b, they end up being at the exact same height.0318
There is no difference: f(a) = f(b); the graph's height does not change in that interval.0324
While the definitions on the previous slide give us formal definitions--they give us something that we can really understand0332
if we want to talk really analytically--we don't really need to talk analytically that often in this course.0338
It is going to be easiest to find these intervals by analyzing the graph of the function.0343
We just look at the graph and say, "Well, when is it going up? When is it going down? And when is it flat?"0347
That is how we will figure out our intervals.0352
We won't necessarily be able to find precise intervals; since we are looking at a graph, we might be off by a decimal place or two.0354
But mostly, we are going to be pretty close; so we can get a really good idea of what these things are--0360
what these intervals of increasing, decreasing, or constant are.0366
So, we get a pretty good approximation by looking at a graph.0370
And if you go on to study calculus, one of the things you will learn is how to find increasing, decreasing, and constant intervals precisely.0373
That is one of the major fields, one of the major uses of calculus.0380
You won't even need to look at a graph; you will be able to do it all from just knowing what the function is.0383
Knowing the function, you will be able to turn that into figuring out when it is increasing, when it is decreasing, and when it is flat.0387
You will even be able to know how fast it is increasing and how fast it is decreasing.0392
So, there is pretty cool stuff in calculus.0395
All right, intervals show x-values: for our intervals of increasing, decreasing, and constant,0399
remember, we are giving intervals in terms of the x-values; it is not, not the points.0403
We describe a function's behavior by saying how it acts within two horizontal locations.0411
We are saying between -5 and -3, horizontally; it is not the point (-5,-3); it is between the locations -5 and -3.0416
And don't forget, we always read from left to right; it is reading from left to right, as we read from -5 up until -3.0425
So, the other thing that we need to be able to do is: we need to always put it in parentheses.0432
Parentheses is how we always talk about increasing, decreasing, and constant intervals.0437
Why do we use parentheses instead of brackets?0442
Well, think about this: a bracket indicates that we are keeping that point;0444
a parenthesis indicates that we are dropping that point, not including that in the interval.0449
But the places where we change over, the very end of an interval, is where we are flipping0453
from either increasing to decreasing or increasing to constant; we are changing from one type of interval to another.0457
So, those end points are going to be changes; they are going to be places where we are changing from one type of interval to another.0463
So, we can't actually include them, because they are switchover points.0470
We want to only have the things that are actually doing what we are talking about.0473
The switchovers will be switching into something new; so we end up using parentheses.0476
All right, a really quick example: if we have f(x) = x^{2} - 2x, that graph on the left,0480
then we see that it is decreasing until it bottoms out here; where does it bottom out?0485
It bottoms out at 1, the horizontal location 1; and it is decreasing all the way from negative infinity, out until it bottoms out at 1.0489
And then, it is increasing after that 1; it just keeps increasing forever and ever and ever.0499
So, it will continue to increase out until infinity.0503
So, parenthesis; -∞ to 1 decreasing; and increasing is (1,∞).0506
We don't actually include the 1, because it is a switchover.0513
At that very instant of the 1, what is it--is it increasing? Is it decreasing?0515
It is flat technically; but we can't really talk about that yet, until we talk about calculus.0520
So for now, we are just not going to talk about those switchovers.0524
All right, the next idea is maximums and minimums.0527
Sometimes we want to talk about the maximum or the minimum of a function, the place where a function attains its highest or lowest value.0530
We call c a maximum if, for all of the x (all of the possible x that can go into the function), f(x) ≤ f(c).0538
That is to say, when we plug in c, it is always going to be bigger than everything else that can come out of that function,0545
or at the very least equal to everything else that can come out of the function.0553
A minimum is the flip of that idea; a minimum is f(c) is going to be smaller or equal to everything else that can be coming out of that function.0556
So, a maximum is the highest location a function can attain, and a minimum is the lowest location a function can attain.0565
On this graph, the function achieves its maximum at x = -2; notice, it has no minimum.0575
So, if we go to -2 and we bring this up, look: the highest point it manages to hit is right here at -2.0580
Why does it not have a minimum? Well, if we were to say any point was its minimum--look, there is another point that goes below it.0589
So, since every point has some point that is even farther below it, there is no actual minimum,0594
because the minimum has to be lower than everything else.0600
There is a maximum, because from this height of 3, we never manage to get any higher than 3, so we have achieved a maximum.0603
And that occurs at x = -2; great.0610
We can also talk about something else; first, let's consider this graph,0615
this monster of a function, -x^{4} + 2x^{3} + 5x^{2} - 5x.0619
Technically, this function only has one maximum; you can only have one maximum,0624
and it is going to be here, because it is the highest point it manages to achieve; it would be x = 2.0630
But it actually has no minimum; why does it not have any minimum?0636
Well, it kind of looks like this is the low point; but over here, we managed to get even lower.0638
Over here, we managed to get even lower; and because it is just going to keep dropping off to the sides,0643
forever and ever and ever, we are going to end up having no minimums whatsoever in this function,0647
because it can always go lower; there is no lowest point it hits; it always keeps digging farther down.0651
But nonetheless, even though there is technically only one maximum and no minimums at all,0659
we can look at this and say, "Well, yes...but even if that is true, that there isn't anything else,0663
this point here is kind of interesting; and this point here is kind of interesting,0669
in that they are high locations and low locations for that area."0674
This is the idea of the relative minimum and maximum; we call such places--these places--0679
the highest or lowest location (I will switch colors...blue...oh my, with yellow, it has managed...blue here; green here)...0686
relative minimums are the ones in green, and the relative maximums are the ones in blue.0701
And sometimes the word "local" is also used instead; so you might hear somebody flip between relative or local, or local or relative.0708
These places are not necessarily a maximum or a minimum for the entire function, for every single place.0715
But they are such a maximum or minimum in their neighborhood; there is some little place around them where they are "king of their hill."0721
So, this one is the maximum in this interval, and this one is the minimum in this interval; and this one is the maximum in this interval.0730
But if we were to look at a different interval, there would be no maximum or minimum in this interval,0737
because it just keeps going down and down and down.0740
And if we were to look at even in here, it is clearly right next to them--if we were to put a neighborhood around this, it would keep going down.0743
It is not the shortest one around; it is not the highest one around.0749
But these places are the highest or lowest in their place.0754
OK, so this gives us the idea of a relative maximum or a relative minimum.0761
Formally, a location, c, on the x-axis is a relative maximum if there is some interval,0765
some little place around that, some ball around that, that will contain c, such that,0770
for all x in that interval, f(x) ≤ f(c)--in its neighborhood, c is the highest thing around.0775
It is greater than all of the other ones.0783
Similarly, for a relative minimum, there is some interval such that f(x) is going to be less than or equal to f(c).0785
In its neighborhood, it is the lowest one around; lowest one around makes you a minimum--highest one around makes you a maximum--0792
that is to say, a relative maximum or a relative minimum.0799
Once again, this is sort of like what we talked about before with the previous formal definition for maximum and minimum,0804
and also for the formal definition of intervals of increasing, decreasing, and constant.0809
Don't get too caught up on what this definition means precisely.0813
The important thing is that we have this graphical picture in our mind that relative maximum just means the high point in that area.0817
And relative minimum just means the low point in that area; that is enough for us to really understand what is going on here.0823
Getting caught up in these precise things is really something for a late, high-level college course to really get worried about.0829
For now, it is enough to just get an idea of "it is the high place" or "it is the low place."0835
Don't forget: the terms relative and local mean basically the same thing--actually, they mean exactly the same thing.0841
They can be used totally interchangeably; and some people prefer to use one; some people prefer to use another.0846
Some people will flip between the two; so don't get confused if you hear one or you hear another one; they just mean the same thing.0851
To distinguish relative local maximums and minimums from a maximum and minimum over the entire function,0859
we can use the terms "absolute" or "global" to denote the latter.0864
If we want to say it is the maximum over the entire function, we could call it the absolute maximum or the global maximum.0868
So, an absolute, global maximum/minimum is where the function is highest/lowest over the entire function,0875
which is exactly how we defined maximum/minimum at first, before we started to talk about the idea of relative maximum/relative minimum.0881
So, absolute or global maximum/minimum is over the entire function--the function's highest/lowest over everywhere in the domain.0886
If we want to talk about all of the relative or absolute maximums/minimums in the functions, we can call the them the extrema (or the "ex-tray-ma").0899
Why? Because they are the function's extreme values: they are the extreme high points0906
and the extreme low points that the function manages to go through, so we can call them the extrema.0911
So, there we are; there is just something for us: extrema.0916
If we want to talk about relative or absolute maximums/minimums in general, we use this word to do it.0919
And absolute or global talks about the single highest or single lowest;0924
relative just talks about one that is high or low in its neighborhood, in the area around that point.0928
Just like find increasing/decreasing/constant intervals, we want to do this from the graph.0936
We don't want to really get too worried or too caught up on these very specific definitions,0941
the formal definitions we were talking about on the previous slide.0945
We just want to say, "OK, yes, we see that that is a high point on the graph; that is a low point on the graph."0948
So, find your minimums; find your maximums by looking at the graph.0953
And once again, if you go on to study calculus, you will learn how to find extrema precisely, without even needing to look at a graph.0957
You will be able to find them exactly; you won't have to be doing approximations because you are looking at a graph.0963
And you won't even have to look at a graph to find them.0966
So once again, calculus is pretty cool stuff.0968
Average rate of change: this also can be called average slope.0972
When we talked about slope in the introductory lessons, we discussed0976
how it can be interpreted as the rate of change, how fast up or down the line is moving.0978
If we have a line like this, it is not moving very fast up; but if we have another line like this, it is moving pretty quickly up.0984
So, it is a rate of change; the slope is how fast it is changing--the rate of change; how fast are we going up?0993
Now, most of the functions we are going to work with aren't lines; but we can still use this idea.0999
We can discuss a function's average rate of change between two points.1003
So, if an imaginary line is drawn between two points on a graph, its slope is the average rate of change.1009
Say we take two points, this point here and this point here; and we draw an imaginary line between them.1015
Then, the slope of that imaginary line is the average rate of change,1022
because what it took to get from this point to the second point is that we had to travel along this way.1025
And while we actually went through this curve here--we actually went through this curve,1031
but on the whole, what we managed to do, on average, is: we really just kind of went along on that line.1037
We could forget about everything we went through, and we could just ask, "Well, what is the average thing that happened between these two points?"1045
And that would be our average rate of change--how fast we were moving up from our first point to our second point.1050
So, if we want to find the average rate of change, how do we do this?1057
Let's say we have two locations, x_{1} and x_{2},1061
and we want to find the slope of that imaginary line between those two points on the function graph.1065
So, that line is sometimes called the secant line; for the most part, you probably won't hear that word too often.1072
But in case it comes up, you know it now.1078
Remember, if we want to find what the slope of this imaginary line is, the slope of this secant line, we know what slope is.1080
How do we find slope? Remember, slope is the rise over the run, so it is the difference between our heights1088
y_{2} and y_{1}, our second height and our first height--what did our height change by,1094
and what did our horizontal location change by--our second location minus our first location?1099
So, our horizontal distance is x_{2} - x_{1}; and our vertical distance is y_{2} - y_{1}.1104
So, y_{1} is the height over here; y_{2} is the height over here.1115
y_{2} - y_{1} over x_{2} - x_{1} is the rise, divided by the run.1119
But what are y_{1} and y_{2}?--if we want to figure out what y_{1} and y_{2} are,1127
well, we just need to look at what x_{1} and x_{2} are.1132
So, since x_{1} and x_{2} are coming to get placed by the function,1134
then y_{2}'s height is really just f(x_{2}), because that is how the graph gets built.1141
The input gets dropped to an output; we map an input to an output.1146
And y_{1} over here is from f(x_{1}).1151
So, since our original slope formula is y_{2} - y_{1} over x_{2} - x_{1},1156
and we know that y_{1} is the same thing as f(x_{1}) and y_{2} is the same thing as f(x_{2}),1160
we can just plug those in, and we get the change in our function outputs, f(x_{2}) - f(x_{1}),1164
divided by our horizontal distance, x_{2} - x_{1}.1171
For our average rate of change, we just look at how much our function changed by between those horizontal locations.1174
How much did its output change by? Divide that by how much our distance changed by.1180
It is often really useful and important to find what inputs cause a function to output 0.1186
So, if we have some function f, we might want to know what we can put into f that will give out 0.1191
That is the values of x such that f(x) = 0.1197
Graphically, since f(x)...remember, f(x) is always the vertical component; the outputs come to the vertical;1200
so, if our outputs are coming from the vertical, then 0 is going to be the x-axis.1208
We have a height of 0 here; so graphically, we see that this is where the function crosses the x-axis.1213
Our crossing of the x-axis is where f(x) = 0.1219
This idea of f(x) = 0 is so important that it is going to go by a bunch of different names.1224
It can be called the zeroes of a function; it can be called the roots of a function; and it can be called the x-intercepts.1229
x-intercepts--that makes sense, because it is where it crosses the x-axis.1236
Zeroes make sense, because it is where we have the zeroes showing up.1241
But how can we remember roots--why is roots coming out?1245
Well, one way to think about it--and actually where this word's origin is coming from--1249
is because it is the roots that the function is planted in.1254
The function we can think of as being planted in the ground (not literally the ground, but we can think of it as being the ground of the x-axis).1258
So, it is like the function has put down roots in the soil.1265
It is not exactly perfect, but that is one good mnemonic to help us remember.1269
"Roots" means where we are stuck in the soil; it is where we are stuck in the x-axis;1273
it is where we have f(x) equal to 0, or where we have an equation equal to 0.1277
But all of these things--zeroes, roots, x-intercepts--they all mean the same thing.1282
They are the x such that f(x) = 0; we can also use these for equations--1287
we might hear it as the zeroes of an equation, the roots of an equation, or the x-intercepts of an equation.1291
There is no one way to find zeroes for all functions.1297
We are going to learn, for some functions, foolproof formulas to find zeroes, to tell us if there are zeroes and what those zeroes are.1301
But for other functions, it can be very difficult--very, very difficult, in fact--to find the zeroes.1308
And although we are going to learn some techniques to help us on the harder ones, there are some that we won't even see1312
in this course, because they are so hard to figure out.1316
But right now, the important thing isn't being able to find them, but just knowing1319
that, when we say zeroes, roots, x-intercepts of a function, or an equation, we are just talking about where f(x) = 0.1323
So, don't get too caught up right now in being able to figure out how to get those x-values such that f(x) = 0.1331
Just really focus on the fact that when we say zeroes, roots, or x-intercepts, all of these equivalent terms,1337
we are just saying where the function is equal to 0--what are the places that will output 0?1344
Even functions: this is a slightly odd idea (that was an accidental joke).1351
Even functions: some functions behave the same whether you look left or right of the y-axis.1357
For example, let's consider f(x) = x^{2}: it is symmetric around the y-axis.1363
What do I mean by this? Well, if we plug in f(-3), that is going to end up being (-3)^{2}, so we get 9.1368
But we could also plug in the opposite version to -3; if we flip to the positive side, -3 flips to positive 3.1375
If we plugged in positive 3, then f(3) is 3^{2}, so we get 9, as well.1383
It turns out that plugging in the negative version of a number or the positive version of a number,1390
-3 or 3, we end up getting the same thing; for -2 and 2, we end up getting the same thing.1395
For -47 and 47, we end up getting the same thing.1400
So, whatever we plug in, as long as they are exact opposites horizontally--1405
they are the same distance from the y-axis--the points are symmetric around the y-axis--1409
they are going to come out to the same height; they are going to have the same output.1414
We call this property even; and I want to point out that it is totally different from being an even number.1419
It is different from an even number--not the same thing as that.1423
But we call this property even for a function.1427
A function is even if all of the x for its domain, for any x that we plug in...1432
if we plug in the negative x, that is the same thing as the positive x.1437
Plugging in f(-x) is equal to plugging in f(x); so we plug in -x into the function, and we get the same thing as if we had plugged in positive x.1442
We can flip the signs, and it won't matter, as long as it is just negative versus positive.1451
Why do we call it even? It has something to do with the fact that all polynomials where all of the exponents1457
end up being even exponents--they end up exhibiting this property.1462
But then, this property can be used on other things; so don't worry too much about where the name is coming from.1465
But just know what the property is: f(-x) = f(x).1470
Odd functions are the reverse of this idea: other functions will behave in the exact reverse.1476
The left side is the exact opposite of the right side; for example, f(x) = x^{3} behaves like this.1481
If we plug in -3, we get -3 cubed, so we get -27; but if we had plugged in positive 3, we would get positive 3 cubed, so we would get positive 27.1488
So, you see, you plug in the negative version of a number, and you plug in the positive version of a number;1500
and you are going to get totally opposite answers.1508
However, they are only flipped by sign; -27 and 27 are still somewhat related.1510
They are very different from one another--they are opposites, in a way; but we can also think of them as being perfect opposites.1515
-27's opposite is positive 27; so an odd function is one that behaves like this everywhere.1520
We call this property odd; it is totally different, once again, from being an odd number.1528
A function is odd if, for all x in its domain, f(-x) is equal to -f(x).1532
And that is a little confusing to read; but what that means is that, if we plug in -x,1538
then that is going to give us the negative version of if we had plugged in positive x.1543
So, if we plug in a negative number, and then we plug in a positive number, the outputs1549
that come out of them will be positive-negative opposites.1555
One of them will be positive; the other one will be negative.1559
So, negative on one side and positive on one side means that the outputs will also be negative on one side and positive on the other side.1561
It is not necessarily going to be the case that the negative side will always put out negative outputs.1568
But it will be the case that it will be flipped if it is odd.1572
This will make a little more sense when we look at some examples.1575
And once again, why are we calling it odd?1577
Once again, don't worry too much about it, but it because it is connected to polynomials where all of the exponents are odd numbers.1579
But don't really worry about it; just know what the property is.1585
Even/odd functions and graphs: we can see these properties in the graphs of functions.1589
An even function is symmetric around the y-axis: it mirrors left/right, because when we plug in a positive number,1593
and we plug in a negative number, as long as they are the same number, they end up getting put to the same location.1600
They get output to the same place.1605
An odd function, on the other hand, is symmetrical around the origin, which means we mirror left/right and up/down,1607
because when we plug in the positive version of a number, it gets flipped to the negative side, but also shows up on the opposite side.1613
It flips to the negative height or the positive height; it flips the positive/negative in terms of height.1622
So, let's look at some examples visually; that will help clear this up.1627
An even one: f(-x) = f(x); let's see how this shows up; if we plug in 0.5, we get here; if we plug in -0.5, we get here.1630
And look, beyond the fact that I am not perfect at drawing, they came out to be the same height.1642
If we plug in 2.0, and we plug in -2.0, they came out to be the same height.1648
You plug in the negative number and the positive number, and they end up coming out to be the same height.1661
That is what it means to be even; and since all of the positives will be the same as the negatives,1666
we end up getting this nice symmetry across the y-axis; it is just a perfect flip.1670
If we took the two halves and folded them up onto each other, they would be exact perfect matches; it is just mirroring the two sides.1676
Odd is sort of the reverse of this: f(-x) = -f(x).1684
For example, let's plug in -1: we plug in -1, and it ends up being at this height, just a little under 2.1690
Let's see what happens when we plug in positive 1; when we plug in positive 1, it ends up being just a little under -2.1699
So, we flip the horizontal location; that causes our vertical location to flip.1708
Let's try another one: we plug in 2.0, and we are practically past it; so we should be just a little bit before 2.0.1713
And we plug in -2.0, once again, just a little past it; so we are just a little before -2.0.1722
And look: we end up being at the same distance from the x-axis, but in totally opposite directions.1727
2.0, positive 2.0, causes us to go to positive 4 in terms of height; but -2.0 causes us to go to -4 in terms of height.1735
So, they are going to flip; if you flip horizontally, you also flip vertically; and that is why we mirror left/right and mirror up/down.1744
We are not just flipping around the y-axis; we are flipping around the origin,1752
because we are flipping the right/left and the up/down; flipping around the origin is flipping the right/left and the up/down.1759
We mirror left/right; we mirror up/down; that is what is happening with an odd function.1771
All right, we are finally ready for some examples.1776
There are a bunch of different properties that we covered; now, let's see them in use.1778
The first example: Using this graph, estimate the intervals where f is increasing and decreasing.1781
Find the locations of any extrema/relative maximums/minimums.1787
And our function is -1.5x^{4} + x^{3} + 4x^{2} + 3.1790
Now, that is just so we can have an idea that that is what that function looks like.1794
But we are not really going to use this thing right here; it is not really going to be that helpful for us figuring it out.1798
So first, let's figure out intervals where f is increasing or decreasing.1802
First, it is increasing from all the way down (and it sounds like we can probably trust1807
that it is going to keep going down, because we have -1.5x^{4}); it is increasing up until...it looks like just after -1.0.1812
It is increasing from negative infinity (because it is going all the way to the left--it is going up1824
as long as we are coming from negative infinity, because it goes down as we go to the left, but we read from left to right),1835
so it is increasing from negative infinity up until...let's say that is -0.9, because it is just after -1.0.1839
And then, it is also going to be increasing from here...let's say it starts there...up until about this point.1846
So, where is that? It is probably about 1.4; so it is increasing from 0 up until 1.4.1852
Where is it decreasing? It is decreasing from this point until this point.1864
That was -0.9 that we said before; so we will go from -0.9 up until 0.1875
And then, it increased up until 1.4; so now it is going to be decreasing from 1.4.1881
And it looks like it is going to just keep going down forever, and it does indeed.1887
So, it is going to be all the way out until infinity; it is going to continue decreasing; great.1890
Now, let's take a look at the extrema; where are the relative maximums/minimums?1895
We have relative maximums/minimums at all of these flipovers that we have talked about, here, here, and here.1902
So, our relative maximum/minimum, our high location, the absolute maximum/minimum, is going to be up here.1909
Relative maximums: we have x =...we said that was 1.4, and that point is going to be 1.4.1916
Let's take a look, according to this...and it looks like it is just a little bit under 8; let's say 7.9.1931
And then, the other one, the lesser of them, but still a relative maximum--it is occurring at x = -0.9.1940
So, its point would be -0.9; and we look on the graph, and it looks like it is somewhere between 4 and 5.1949
It looks a hair closer to 5, so let's say 4.6; great.1958
Relative minimum--our low place: well, we can be absolutely sure of what the x is there--it is pretty clear that that is x = 0.1962
And what is the height that it is at right there? It looks like it is exactly on top of the 3, so it is (0,3).1972
We have all of the intervals of increasing and decreasing.1978
And we also have all of our extrema, all of our relative maximums and minimums; great.1981
Example 2: A ball is thrown up in the air, and its position in meters is described by location of t.1987
Distance of t is equal to -4.9t^{2} + 10t, where t is in seconds.1993
OK, so we have some function that describes the height of the ball--where the ball is.1998
What is the ball's average velocity (speed) between 0 seconds and 1 second,2004
between 0 and 0.01 seconds, and between 0 and 2.041 seconds?2007
OK, at first, we have some idea...if we were to figure out what this function looks like, it is a parabola.2012
It has a negative here, so it is ultimately going to go down.2017
And it has the 10t here; if we were to graph it, it would look something like this.2020
And that makes a lot of sense, because if we throw a ball up, with time, the ball is going to go up and them come back down.2024
So, that seems pretty reasonable: a ball is thrown up in the air, and its position is given by this.2031
But how does speed connect to position? Well, we think, "What is the definition of speed?"2036
We don't exactly know what velocity is, necessarily; maybe we haven't taken a physics course.2040
But we probably know what speed is from before in various things.2044
Speed is distance divided by time, so distance over time equals speed.2047
It seems pretty reasonable that velocity is going to be the same thing.2058
That is not exactly true, if you have actually taken a physics course; but that is actually going to work on this problem.2063
We are going to have a good idea of what is going on with saying that that is true.2068
All right, so what is the ball's average velocity?2072
The average velocity is going to be the difference in its height, divided by the time that it took to make that difference in height.2074
So, we are going to be looking for distance.2080
If we have 2 times, time t_{1} to time t_{2}, it is going to be the location at time t_{2},2084
minus the location at time t_{1}, over the difference in the time, t_{2} - t_{1}.2095
Oh, and that makes a lot of sense; it is going to be connected, probably, to what we learned in this lesson,2102
since with student logic, they normally try to give us problems that are going to be based off of what we just learned.2107
So, t_{2} - t_{1}...this looks just like average rate of change.2112
The average rate of something's position--that would make sense, that how fast it is going is the rate of change; the thing is changing its location.2116
The rate at which you are changing your location is the velocity that you have; perfect.2124
Great; so we need to figure out what it is at 0 seconds and what it is at 1 second right away.2128
So, the location at 0 seconds; we plug that in...-4.9(0)^{2}, plus 10(0)...that is just 0, which makes a lot of sense.2133
If we throw a ball up, at the very beginning it is going to be right at the height of the ground.2140
Distance at time 1 is going to be -4.9 times 1^{2}, plus 10 times 1; so we get 5.1.2146
If we want to figure out what is its average velocity between 0 seconds and 1 second, then we have d(1) - d(0)/(1 - 0), equals 5.1 - 0/1, which equals 5.1.2159
What are our units? Well, we had distance in meters, and time in seconds; so meters divided by seconds...we get meters per second.2180
That makes sense as a thing to measure velocity and speed.2189
All right, next let's look at between 0 and 0.1 seconds.2192
If we want to find 0.01 seconds, the location at 0.01 equals -4.9(0.01)^{2} + 10(0.01).2196
Plug that into a calculator, and that is going to end up coming out to be 0.09951; so let's just round that up2211
to the much-more-reasonable-to-work-with 0.01.2218
OK, so it rounds approximately to 0.01; so let's see what is the average rate of change.2222
The average rate of change, then, between 0 and 0.01 seconds, is going to be d(0.01) - d(0) over 0.01 - 0.2228
That equals...oops, sorry, my mistake: 0.01 is not actually what it came out to be when we put it in the calculator.2245
I mis-rounded that just now; it was 0.09951, so if it is 0.09951, if we are going to round that2253
to the much-more reasonable-to-work-with thing, we actually get approximately 0.1.2265
So, it is not 0.01; 0.01 is still on the bottom, but the top is going to end up coming out to be 0.1 - 0, divided by 0.01; sorry about that.2271
It is important to be careful with your rounding.2288
That comes out to be 0.1 over 0.01, which comes out to be 10 meters per second.2290
And now, you probably haven't taken physics by this point; but if you had, you would actually know that -4.9t^{2}...2298
that is the thing that says the amount that gravity affects where its location is.2307
The 10t is the amount of the starting velocity of the ball.2312
The ball gets thrown up at 10 meters per second, so it makes sense that its average speed2315
between 0 and 0.01--hardly any time to have changed its speed--is going to be pretty much what its speed started at.2320
That 10 meters per second is actually showing up there.2327
So, there is a connection here between understanding what the physics going on is and the math that is connecting to it.2329
All right, finally, between 0 and 2.041 seconds...let's plug in d(2.041) = -4.9(2.041)^{2} + 10(2.041).2335
So, that is going to come out to be -0.0018; so it seems pretty reasonable to just round that to a simple 0.2354
Now, what does that mean? That means, at the moment, 2.041 seconds--that is when the ball hits the ground.2365
It goes up at 0, and then it comes back down.2371
And at 2.041 seconds after having been thrown up, it hits the ground precisely at 2.041 seconds.2374
So, 2.041 seconds--then it has a 0 height; so what is its average velocity between 0 and 2.041 seconds?2380
Location at 2.041 minus location at 0, divided by 2.041 - 0, equals 0 minus 0, over 2.041, which equals 0 meters per second,2388
which makes sense: if we throw the ball up, and then we look at the time when it hits the ground again,2404
well, on average, since it went up and it went down, it had no velocity,2410
because the amount of time that it has positive velocity going up and the amount of time that it has negative velocity going down--2415
it has cancelled itself out, because on average, between the time of its starting on the ground2420
and ending on the ground, it didn't go anywhere.2425
So, on average, its velocity is 0, because it didn't make any change in its location; great.2427
The next example--Example 3: Find the zeroes of f(x) = 3 - |x + 3|.2433
Remember: zeroes just mean when f(x) = 0; so we can just plug in 0 = 3 - |x + 3|.2439
So, we have |x + 3| = 3; we just add the absolute value of x + 3 to both sides.2451
We have |x + 3| = 3; that is what we want to know to figure out when the zeroes are.2458
When is this true? Remember, absolute value of -2 is equal to 2, which is also equal to the absolute value of positive 2.2462
So, the absolute value of x + 3...we know that, inside of it, since there is a 3 over here...2474
there could be a 3, or there could be a -3.2480
So, inside of that absolute value, because we know it is equal to 3, we know that there has to currently be a 3, or there has to be a -3.2484
We aren't sure which one, though; so we split it into two different worlds.2498
We split it into the world where there is a positive on the inside, and we split it into the world where there is a negative on the inside.2502
In the positive world, we know that what is inside, the x + 3, is equal to a positive 3.2508
In the negative world, we know that the x + 3 is equal to a negative 3.2518
Now, it could be either one of these; either one of these would be true; either one of these would produce a 0 for the function.2523
So, let's solve both of them: we subtract by 3 on both sides over here; we get x = 0.2529
We subtract by 3 on both sides over here; we get x = -6.2535
So, the two answers for the roots are going to be -6 and 0; that is when the zeroes of f(x) show up.2538
The zeroes of f(x) are going to be at x = 0 and x = -6.2546
And if we plug either one of those into that function, we will get 0 out of the function.2549
The final example: Show that x^{6} - 4x^{2} + 7 is even;2556
show that -x^{5} + 2x^{3} - x is odd; and show that x + 2 is neither.2559
All right, the first thing we want to do is remind ourselves of what it means to be even.2564
To be even means that when we plug in the negative version of a number, a -x is the same thing as if we had plugged in the positive x.2572
It doesn't have any effect.2581
And the odd version...actually, let's put it in a different color, so we can see how all of the problems match up to each other.2583
If we do with the odd version, then if we plug in the negative of a number,2590
it comes out to be the negative of if we had plugged in the positive version of the number.2595
All right, so the first one: Show that x^{6} - 4x^{2} + 7 is even.2599
So, that was really seeing that expression as if it were a function; so let's show this2605
by showing that if we plug in -x, it is the same thing as if we plug in positive x.2609
On the left, we will plug in -x; -x gets plugged in; it becomes (-x)^{6} - 4(-x)^{2} + 7 =...2613
if we plugged in just plain x, we would have plain x^{6} - 4x^{2} + 7; great.2622
(-x)^{6}...remember, a negative times a negative cancels out to a positive.2630
We have a 6 up here; we are raising it to the sixth power, so we have an even number of negatives.2634
Negative and negative cancel; negative and negative cancel; negative and negative cancel.2642
That is a total of 6 negatives; they all cancel each other out; so we actually have (-x)^{6} being the same thing as if we just said x^{6}.2645
Minus 4...the same thing here: -x times -x cancels and just becomes plain x^{2}...plus 7 equals2652
x^{6} - 4x^{2} + 7; it turns out that it has no effect.2660
If we plug in a negative x, we get the same thing as if we had plugged in the positive x.2666
Plugging in a negative version of a number is the same thing as plugging in the positive version of the number.2669
So, it checks out; it is even; great.2672
The next one: let's look at odd: -x^{5} + 2x^{3} - x is odd.2677
We will do the same sort of thing: we will plug -x's in on the left side.2682
-(-x)^{5} + 2(-x)^{3} - (-x); what is going to go on the right side?2686
Well, remember: if we plug in the negative version of the number, then it is the negative of if we plugged in the positive version of the number.2696
So, it is the negative of if we had plugged in the positive version of the number.2703
Plugging in the positive version of the number is just if we have the normal x going in: -x^{5} + 2x^{3} - x.2706
All right, so -(-x)^{5}: well, what happens when we have (-x)^{5}--what happens to that negative?2714
Negative and negative cancel; negative and negative cancel; negative--that fifth one, because it is odd, gets left over.2720
So, we have negative; and we just pull that negative out--it is the same thing as -x^{5}.2727
Plus 2...once again, it is odd; a negative and a negative cancel; we are left with one more negative, for a total of 3 negatives; we are left with a negative.2732
So, we get 2(-x)^{3} minus...we can pull that negative out, as well...-x...equals...2739
let's distribute this negative; so we get...distribute...cancellation...a negative shows up here...cancel;2747
we get positive x^{5} minus 2x^{3} + x.2754
So, let's finish up this left side and do cancellations over here as well; positive, positive; this stays negative.2761
Positive, positive; so we get x^{5} - 2x^{3} + x equals the exact same thing over here on the right side.2767
It checks out; yes, it is odd; great.2778
Finally, let's show that x + 2 is neither; so, to be neither, we have to fail at being this and fail at being this.2782
So, to be neither, it needs to fail being odd and being even; it needs to fail even and odd.2791
Let's just try plugging in a number; let's try plugging in, say, -2.2809
If we look at x = -2, then that would get us -2 + 2, which equals 0.2814
Now, what if we plugged in the flip of -2--we plugged in positive 2?2821
x = positive 2...we plug that into x + 2, and we will get 2 + 2, which equals 4.2825
Now, notice: 0 is not equal to 4; we just failed being even up here,2832
because the negative number and the positive version of that number don't produce the same output.2841
Plug in -2; you get 0; plug in +2; you get 4; those are totally different things, so we just failed to be even; great.2846
Next, we want to show that it is not odd.2854
Odd was the property that, if we plug in the negative, it is going to be equal to the negative of the positive one.2857
So, 0 is not equal to -4 either, right? If we plug in -2, we get 0, and if we plug in positive 2,2861
it turns out that that is not -0, or just 0; it turns out that that is 4.2869
So, we fail to be odd as well, because it isn't the case that if we plug in opposite positive/negative numbers,2873
we don't get opposite positive/negative results, because 0 is not the opposite of -4; it is just the opposite of 0, so it fails there.2881
So, it checks out: that one is neither; great.2890
All right, we just learned a whole bunch of different properties; and they will each come up in different places at different times.2896
Just remember these: keep them in the back of your mind.2901
If you ever need a reminder, come back to this lesson and just refresh what that one meant,2903
because they will show up in random places; but they are all really useful.2907
And we will see them a lot more as we start getting into calculus.2910
Once you actually get to calculus, this stuff, especially the stuff at the beginning of this,2913
where we talked about increasing and decreasing and relative maximums and minimums--2915
that stuff is going to become so important if you are going to understand why we are talking about it so much right now in this course.2919
All right, I hope you understood everything; I hope you enjoyed it; and we will see you at Educator.com later--goodbye!2924
1 answer
Last reply by: Professor Selhorst-Jones
Sun Aug 11, 2013 11:03 PM
Post by Tami Cummins on August 11, 2013
In the second part of example 2 what about the negative?
1 answer
Last reply by: Professor Selhorst-Jones
Thu Jul 11, 2013 1:36 PM
Post by Montgomery Childs on June 26, 2013
Dear Mr. Jones,
I really appreciate the time you spend on "definitions" of math terms - i have come to realize this is one of the biggest issues i have had over the years - not the math. This helps so much in my understanding of relationships! Very cool!!!