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Welcome. In this unit, we'll be reviewing
your algebra and geometry skills. This are
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skills you probably learned before in your
high school or previous mathematics
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courses. But these will be specifically
focusing on things that you'll need to
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focus on your precalculus, and ultimately,
your calculus classes. Let's begin by
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looking at an example of when you might
need some algebra or geometry skills. Back
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in the 70s, there was a really popular
Tootsie pop commercial featuring an owl
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who ask the question, how many licks does
it take to get to the Tootsie Roll center
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of a Tootsie pop? For those of you who
aren't familiar with it, Tootsie pops were
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a little candy with a Tootsie Roll center
and some coating on the outside. I brought
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my Tootsie Pop today. Well, this isn't
actually a Tootsie Pop, it's a little bit
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larger, but we could ask the question, how
many licks would I have to make on this
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lollipop to get to the candy in the
center? So, this is basically a math
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question, although, it seems like a candy
question. We could do an experiment.
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Suppose I took this Tootsie Pop, unwrapped
it, and suppose I licked it for a minute.
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And then measured the circumference to
figure out how much of the lollipop was I
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able to lick off in a minute, and suppose
I repeat that experiment several times. So
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I lick it for one minute, then I lick it
for another minute, and I get some data.
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Let's look at the data when I licked this
lollipop. On this graph here, you see some
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blue dots. Those blue dots represent the
circumference of various points in time,
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during my experiement. As I'm licking this
lollipop, I'm keeping track of the
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circumference, at various measurements, at
various points in time, and I'm denoting
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by a blue dot. You notice this data looks
like it's sort of straight. If I was going
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to draw a straight line through the data
most of the data points would be pretty
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close to that straight line. Let's look at
that. Here I've drawn a black line that
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pretty well fits this data. This is called
the line of best fit, and it can be found
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using a technique called linear
regression. On there, I also have an
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equation for the data. That black equation
y = -0.34x + 6.34 is the equation of the
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line of best fit. It's basically an
equation for the line that goes through
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the data as close as possible. Well, let's
talk more about what this equation
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represents and what sort of things we can
do with it. Namely, we wanna answer that
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question from the old Tootsie Pop
commercial, how many licks does it take to
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finish the lollipop? Well, let's take a
look at the equation, the equation that
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best fit my data from my experiment was, y
= -0.34x plus 6.34. Let's pick this
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equation apart a little bit. The x in the
equation corresponded to time, that was
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the x-axis or the lower axis on my figure.
It basically just corresponded to
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different values of time for licking the
lollipop. The y-axis on my diagram
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corresponded to circumference. So the y
value is a circumference in inches of what
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the current circumference is, at a given
time. Normally, when we look at linear
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equations, we put them in what we call a
standard form or one of several standard
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forms. In particular, my favorite standard
form is the point the slope intercept form
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in which you have both the slope
information of the graph or how steep that
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line is, and the intercept. Where does it
hit the y-axis? We usually denote that as,
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y = mx + b. The m corresponds to a slope,
the b corresponds to the y intercept. In
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this case, our slope value is -0.34, the
units on that would be inches per minute.
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Basically, the slope corresponds to how
much of this lollipop, how much of the
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circumference am I licking off in each
minute? And the y intercept, the b value
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of 6.34, what would that correspond to?
Well, that measurement is in inches and
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that corresponds to the initial
circumference of my lollipop, so in this
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particular experiment, I had a lollipop
that initially had a 6.34 inch
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circumference. Well, we still haven't
answered the owl's question from that
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commercial. How many licks does it take to
finish the lollipop? What information
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would we be wanting to look for in order
to answer that question? Well, that's
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really asking us to find the x intercept.
We wanna know what is the time value or
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how long is the time, which corresponds to
x, at which I'd finish the lollipop?
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Finishing the lollipop corresponds to y =
zero.
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Y was our circumference value and when I'm
done with the lollipop, I should have no
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more left, so I'd have a circumference as
zero.
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In order to answer this question then, I
found my data, I found the line which best
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fit my data, and now I want to take my
linear equation and solve it for the x
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intercept. So I plug in y = zero to my
equation. I move all the x terms to one
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side, all the constant terms to the other
side and I can solve for x. And if you see
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by my arithmetic up here, I've gotten that
it would take me 18.7 minutes to finish my
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lollipop, given the data that I've had.
Obviously, my data wasn't perfect. It's
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not exactly a linear function. But in the
real world, we very often approximate real
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world experiments or phenomenon with some
sort of function. In this case, a linear
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function seemed like a good approximation
to my data. Well, let's talk more about,
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what sort of skills will you need to have
in order to use algebra and geometry to
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study problems in precalculus and
calculus. We're gonna be dealing in this
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unit with four main types of functions.
Those four main types are polynomials,
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rational functions, radical functions, and
problems or functions involving absolute
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value. Polynomials you can think of simply
as, sums of powers of x. For example, x^2,
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x^3, x^4, etcetera. Anytime I have a
function that's a sum of those type of
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terms where the base is a variable,
usually x but it could be something else,
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and the power as an integer like two,
three, four, five those we call polynomial
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expressions. The formula I've written down
here for polynomial, you notice it has
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some strange things, an, an - one,
etcetera. Those a's just there taking the
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place of any constant coefficient, just
any real number, just a number there is,
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and that's gonna give us a polynomial. We
also will take about rational functions.
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Rational functions are functions made up
of a numerator and a denominator where
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both the numerator and denominator or
polynomials and I just told you what a
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polynomial was. The one stipulation is, in
mathematics, we always have to be careful
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that we don't wanna divide by zero. So
with our rational functions, we'll usually
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make the stipulation that the denominator
can't equal zero, the denominator equaling
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zero is not part of our domain. There's
two other types of functions you'll be
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talking about in this unit and that one of
those is radical functions. Radical
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functions are things involving square
roots or other type of roots, such as
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third roots, fourth roots, fifth roots,
etcetera. When you deal with these, we
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usually denote it by that square root sign
that you see in the diagram up there, and
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you'll realize when, you'll know when to
recognize those, cuz you'll just see that
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radical form. Finally, the last function
that tends to scare students a little bit,
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but really isn't that scary at all is the
absolute value function. The absolute
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value, simply put, is just the positive
version of the number. If I put in three,
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the positive version of three is still
three.
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However if you put in -three, we want to
switch the sign to make the positive
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version so -three, the absolute value of
negative is simply three.
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The equation that I've written there is
the definition of absolute value. It
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simply says that if x < zero, we have to
make the positive version so I have to
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change the sign or take -x. If x was
already positive, I don't need to do
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anything when I take the absolute value of
x, I simply keep the value x alone. This
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kind of piece-wise define function with
two parts is one that you'll encounter a
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lot in this course. So finally, let's talk
about, what are you gonna learn in this
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unit? In this unit, first and foremost,
you'll learn to solve linear equations
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just like we did with that lollipop
problem. You'll also learn how to solve
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linear inequalities. Inequalit y is like
an equation, except it will either have a
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>, <, or the greater than or equal to or
less than or equal to sign in the
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equation. For example, here I've shown you
a compound inequality. We know that 5x +
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seven < eighteen, but it's also greater
than or equal to three.
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You'll learn how to solve things like
that, find out what values of x makes this
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expression true. You'll also be learning
how to solve inequalities involving
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absolute values. For example, if I tell
you that the absolute value of x - 3x is
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smaller than or equal to two, you wanna
find out what values of x make that
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expression true. We'll also learn how to
factor polynomials. This is one of those
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skills that a lot of students tend to
remember from high school, but, we'll give
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you a refresher and remind you of those
special cases that you may not always
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remember. Here's an example of a
polynomial x^2 + x - two, which I have
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factored or broken into the multiplication
of two linear terms, (x+2) (x-1).
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Another skill you'll be learning how to do
is how to simplify and solve rational
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expressions. So, for example, I've given
you a rational expression using an
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equality and you'll learn how to solve
what values of x make this rational
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expression or ratio of two polynomials
greater than or equal to zero, also known
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as positive. Finally, in this unit you'll
also be working with simplifying and
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solving expressions involving radicals. So
we'll be learning the rules of how to
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manipulate terms within radicals to
simplify the overall expression and make
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it easier to solve certain types of
equations. What are the applications that
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you'll be looking at for algebra and
geometry? Well, there's a lot of them.
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Basically, this section is really focused
on giving you some tools and techniques to
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solve applications you'll be encountering
in later sections. So, for this particular
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section, we won't be focused as much on
what the applications are, but on getting
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you some tools to solve later
applications. Almost any problem I can
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think of in math, science, engineering,
business, is going to involve some sort of
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polynomials, or rational functions, or
absolute value functions. So, these tools
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will have a wide reaching effect. When do
you use algebra? Well, algebra is going to
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come up pretty much any time you have some
quantity you don't know, some other ones
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you do know, and you know a relationship
between the knowns and the unknowns.
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Basically, you're using some known
information to solve for the piece you
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don't know and that's kind of a simplified
definition of what algebra really is.
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Well, thank you and I really hope you
enjoy reviewing your algebra and geometry
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skills. Thank you and I'll see you next
time.