Hello again, Mr. Tarrou. I want to cover one more topic before I head home this afternoon.
You might notice you keep seeing the same shirt in some of these videos. I want to talk
about how do we read graphs, analyze things like domain and range, relative max and mins,
how do I describe where a graph is constant... increasing... or decreasing, and really just
pick this graph apart. Let's talk first about roots. Where are the roots, or the zeros,
or my particular book uses this term...where are the zero's at, or solutions. Where are
these? This all says the same thing by the way. Where are the roots, zeros, or the solutions
of this graph. You find solutions all the time. Every time you solve an equation, use
the quadratic formula, use synthetic division if you are looking for solutions, factoring.
What do you find when you find a solution to an equation? Well graphically what you
are finding is where the graph crosses the x axis. This graph crosses the x axis at negative
four, one, and positive four. So the solutions, the zeros, and the roots... well that is it.
It is x equals -4, 1, and 4. These are the solutions, the zeros, the roots of this graph.
If you have a TI-83 or TI-84 and you want to find an x intercept, are are going to look
for a Zero. Where is the y intercept? Well, the y intercept is where it crosses the y
axis. That is a value of 3. So the y intercept is either 3 or you textbook might write it
as (0,3) if you are naming it as a coordinate. Let's talk about something that my students
have found to be a little bit more confusing. Where is the graph increasing? Where is the
graph, I read graphs left to right... and so should you, as you read the graph left
to right where is the graph going up? Now I know going up you might go up here. Well
a lot of kids will say that it increases to four. Well, the highest y value here is four.
We will talk about that in a minute when I do relative maximum. But, when you are asked
where is the graph increasing, that means as I uncover this graph if you see the graph
going up then the graph is increasing. Then I want the x values. Along the x axis where
is the graph going up? Well it is way down here and it is coming up, and coming, and
coming up as I read left to right... and it stops going up when I get to negative one.
I am going to write my answers in interval notation since I am enforcing that in my PreCalculus
class. But it... So it might look a little bit funny. But it is increasing until you
get to an x value of negative one. So it is increasing from negative infinity, that is
all these negative numbers going to the left forever, up until you get to negative one.
Then it is stops increasing and starts to decrease. But I want increasing and this is
not the only place where the graph increases. It also increases from two to 3, 4, 5, and
6. It starts to increase this bottom, this relative minimum, it starts to increase until
I get up here to... and I don't want to say that y value because I don't want that to
stick in your head... it increases from 2 to 6. So it increases from negative infinity
to one, or from positive two... let me make sure that I counted that right... 1, 2, 3,
4, 5, six. Good. This is along the x axis where you see the graph going up. Let's highlight
the decreasing in a different color. Where is the graph decreasing? Well as you read
the graph left to right, where do you see the graph going down. Again I do not want,
and the textbooks, will not want a y value answer. They want to along the x axis, where
is the graph falling. It reaches the top at the x value of -1 and it falls until you get
to that x value of 2 that we just used for the increasing part because after it falls,
it rises. So it decreases from negative one to positive two. Good! I am going to kind
of keep scrolling my answers down as I go. What about this idea of constant which you
might have heard about? Where is the graph constant? Where is the graph, as you read
it left to right, not changing? Well there it is right there. Along these values the
graph, the x is increasing by the y values are staying the same. That means the graph
is constant. So the graph is constant from an x value of 6 and then it goes to the right
forever. So if you are using inequality notation that means it is constant were x is greater
than 6. In interval notation it starts at 6 it goes to infinity...AND BEYOND. Ok, any
way:) What about the top of the graph? Like this is kind of rising and then falling. So
if you rise and fall you have made a peak in your graph. We call that a relative minimum.
If it were never to go up again, we would call that an absolute maximum. But we are
going to just use relative maximum. Did I just say minimum? Anyway. That is the top
so it is a maximum. Ok. So where is my relative maximum point? My relative maximum occurs
at... or is at (-1,4). If you have a textbook really encouraging and emphasizing function
notation you may see that as f(-1) equals 4. This says that your relative maximum of
4 is occurring at the x value of -1. So use whatever your teacher wants for these answers.
Seven, where is the relative minimum? Where is the bottom of the curve? Is there one even
made? If this curve were to raise up and make a peak and then fall down forever like an
upside down parabola there would be no relative minimum. But I do have one in this graph and
it is right here. It is falling down until you get to a y value of -2 at 2. So we have
a relative minimum, the slope goes from negative to positive, at the point (2,-2). So we have
a relative minimum at (2,-2) or again emphasizing the function notation the function has a relative
minimum of -2 at the x value of 2. Again just use whatever notation your teacher is looking
for in this graph. Domain and range. Well the domain of a graph, one more time if you
have watched some of the other videos, the domain of the graph is what x values does
the graph include. Well this graph comes way from the left, does a little squiggle, and
then keeps shooting over to the right. So this has an x value of 1, 2, 3, 4, 5, 6, negative
seven. This point has an x value of -6. This point has an x value of -5... -4. This point
has an x value of -1. Here it has an x value of 0 being used. This graph contains and uses
along its points all of the x's. So the domain is going to be your x values from negative
infinity to positive infinity, or if your book is not using interval notation you would
say that the domain is all real numbers. I will not because for my kids I am trying to
enforce the use of interval notation. What is the range of this graph? Just picking this
apart. Same old picture. What y values are included along this graph? We have really
small y values. They are coming up from negative infinity. They come up to four, you drop back
down again so I am using some of these y values again... twice... actually even three times.
It falls down again, rises up, and then it stops and it goes flat. So we have y values
of 1, 2, 3, 4, 5... We are using y values of -6, -10, -20, negative infinity. We have
y values of negative three, -2, -1, we have y values of zero... You get my point here?
All of these points are using different y values from negative infinity all the way
up until we get to a y value of 1, 2, 3, 4, 5. We are using the highest y value of five.
So the range of this graph, I am little bit tired with this being the end of the day....
I hope I did not say the wrong word there for a second, but the range of this graph
goes from... well it is y values... and it goes from negative infinity up until you get
to a y value of five. So the range is negative infinity to five. Now for those of you in
my PreCalc class and anyone else using interval notation, that 5 is a concrete value. It does
get used. This is going right along the line y=5. So use that square bracket to indicate
that it is actually being used. So that is your range. If you are using interval notation,
excuse me inequality notation it would be y is less than or equal to 5. I think I have
a little bit of time left just to make a slight change to this graph and explain what this
square bracket stuff is about. If my, this would change some of my previous answers so
keep in mind I just changed my graph so forget all of those previous answers for a second.
If there was a horizontal asymptote and that horizontal asymptote were y=5, and I draw
this graph so that it comes up and approaches five from the bottom but never actually becomes
five. First of all, now my graph is constant nowhere. So that previous answer would be
out the window. And the graph comes up and approaches but does not quite use the y value
of 5. My range would go from negative infinity and then instead of actually using the value
of 5, the graph would just approach the value of 5 and you would not actually use it. So
there you go. BAM!!!