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