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• Hi, I’m Rob. Welcome to Math Antics!

• In this lesson, were gonna learn something that’s an important foundation for tons of math problems,

• including those youll encounter while learning basic Algebra.

• Were gonna learn about graphing,

• which basically means taking mathematical relationships and turning them into pictures.

• Hey Friends, welcome back to The Joy of Graphing.

• Were gonna pick up right where we left off.

• We already have this nice, beautiful function right here.

• But it needs a friend.

• And were gonna do that by adding some points.

• So let’s put the next point right here.

• Now all we need to do is connect those pointscuz theyre all friends.

• And what to friends do?

• They stay connected.

• Oh and look at that

• That’s beautiful.

• Well, not the kind of picture that you’d hang on your wall.

• Graphing just means making a visual representation of an equation or data set so you can understand it better.

• It’s a way of helping you literally SEE how math works.

• When math is just a bunch of numbers and symbols on a page, it can be pretty abstract and hard to relate to.

• But graphing is like a window into the abstract world of math that helps us see it more clearly.

• In fact, the focus of our lesson today actually looks a bit like a window.

• and it’s calledThe Coordinate Plane”.

• The coordinate plane is the platform or stage that our graphing will take place on.

• But to understand how it works, we first need to start with its closest relative; The Number Line.

• You remember how a number line works, right?

• A number line starts at zero, and represents positive numbers as you move to the right

• and negative numbers as you more to the left.

• And there’s usually marks showing where each integer is along the way.

• Now imagine cloning that number line and rotating the copy counter-clockwise by 90 degrees

• so that the second number line is perpendicular to the first and they intersect at their zero points.

• What we have now is a “number plane”. It’s basically like a 2-Dimensional version of a number line,

• but that second dimension makes it much more useful.

• With a simple 1-Dimensional number line,

• we could show where various numbers were located along that line by drawing (or plotting) points.

• But no matter how many points we plot, theyre always on the same line.

• But with a 2-Dimensional number plane, we can plot points anywhere in that 2D area,

• and that opens up a whole new world of possibilities.

• With a 1-Dimensional number line, plotting points was easy.

• You just needed one number to tell you where to plot a point.

• But with a 2-Dimensional number plane, you actually need TWO numbers to plot each point.

• These two numbers are calledcoordinatesbecause theyre the samerankororder

• and they work together to specify the locations of a point on the number plane.

• In fact, that’s why the number plane is often referred to as thecoordinate plane”.

• It’s the stage for plotting coordinates.

• The two numbers are put inside parentheses with a comma between them as a separator.

• So when you see 2 comma 5

• or negative 7 comma 3

• or 0 comma 1.5

• you know youre dealing with coordinates.

• Okay, to understand how coordinates work,

• remember that our number plane is formed by combining two perpendicular number lines.

• From now on, were going to refer to each one of these number lines as anaxis”.

• One of the axes is horizontal (like the horizon)

• which means the other axis is vertical (or straight up and down).

• And theyre often called thehorizontalandverticalaxes because of that.

• But even more often, the axes are referred to by variable-based names.

• The horizontal axis is calledthe X-axisand the vertical axis is calledthe Y-Axis”.

• Why use variable names?

• Well there’s two good reasons.

• The first is that variable names are more flexible than horizontal and vertical,

• which relate to a specific orientation in space that may not always be relevant.

• And the second reason is that each of the two coordinate numbers is actually a variable

• that relates to a specific position along one of the two axes of the coordinate plane.

• And since those variables are usually called ‘X’ and ‘Y’, it makes sense to name the two axes the same way.

• The first coordinate number listed will be called ‘X’

• and the second coordinate number listed will be called ‘Y’.

• And were always going to list the numbers in that same order, ’X’ first and then ‘Y’,

• so that we never get confused about which is which.

• In fact, coordinates are often calledordered pairsbecause

• theyre a pair of numbers that are always listed in the same order:

• ‘X’ value first… ‘Y’ value second.

• So if you have the coordinates, (3, 5) that means X = 3 and Y = 5.

• Pretty easy, right?

• But now how do we actually plot these coordinates (or ordered pairs) on the coordinate plane?

• Well, the first number in the ordered pair tells you where along the X-Axis the point is located,

• and the second number in the ordered pair tells you where along the Y-Axis the point is located.

• The two numbers in an ordered pair work together to define a single point,

• and each one of the numbers only gives you half of the information about where that point is.

• To see how this works, let’s plot the coordinates (3, 2)

• First, we locate the X value along the X-axis, which is at 3 in this case.

• But instead of putting a point there,

• we draw (or just imagine) a line perpendicular to the X-axis that goes through the 3.

• We do that because the first number in the ordered pair only tells us where along the X-axis the point is,

• but it could be ANYWHERE along the Y-axis.

• We won’t know that until we plot the second number.

• So temporarily, we just draw a line there to represent every possible point that could have an X value of 3.

• Next, we locate the Y value along the Y-axis, which is at 2 in this case.

• But again, instead of putting a point there,

• we draw (or just imagine) a line perpendicular to the Y-axis that goes through the 2.

• We do that because the second number in the ordered pair only tells us where along the Y-axis the point is,

• but it could be ANYWHERE along the X-axis.

• So we just draw a line there to represent every possible point that could have a Y value of ‘2’

• Ah, but look what weve got now.

• The first line represents all the possible locations where X equals 3.

• And the second line represents all the possible locations where Y equals 2.

• And the exact point where the two lines intersect represents

• the only point in the entire coordinate plane where both X = 3 and Y = 2.

• That intersections is the location of our point.

• Pretty cool, huh? And that’s a really good way to understand how the coordinate plane works.

• But I want to show you an even easier (and more intuitive) way to actually plot points.

• This more intuitive way involves starting with a point at the origin of the coordinate plane

• and then treating the coordinates like a set of simple instructions

• that tell you how far to move your point in the X and Y directions.

• For example, to plot the coordinates (3, 2) like before, we start by imagining a point at the origin (0, 0)

• Then, we look at the first number in our ordered pair to see how far we need to move our point in the X direction.

• Since X is positive 3, we move our point a distance of 3 units in the positive X direction.

• And then from there, since Y is positive 2, we move our point a distance of 2 units in the positive Y direction.

• So that’s a pretty easy method for plotting points!

• Let’s try it a few more times so you get the idea.

• Let’s plot the coordinates (-4, 3).

• Again, we start by imagining a point at the origin

• and then let the coordinates tell us how far to move it along the X and Y axes.

• Since X is negative 4, we move the point a distance of 4 units,

• but this time in the negative X direction which is to the left.

• And then, since Y is positive 3, we move the point a distance of 3 units in the positive Y direction.

• Now let’s plot the coordinates (-3, -3).

• In this case, X and Y are both negative, so starting with a point at (0, 0)

• we move it 3 units in the negative X direction,

• and then 3 units in the negative Y direction.

• And last, let’s plot the coordinates (4, -2.5).

• Starting at (0, 0) we move the point 4 units in the positive X direction

• and then 2 and 1/2 units in the negative Y direction.

• Okay, so weve plotted four ordered pairs the easy way,

• and did you notice that each of these points is located in a different region of the coordinate place.

• These four regions are calledQuadrantsand their boundaries are defined by the two axes of the coordinate plane.

• The quadrants are named 1 through 4 so we can easily refer to them in conversations if we need to.

• and it contains all of the points where both the X and Y values are positive.

• Quadrant 2 is the upper left,

• and it contains all of the points that have a negative X value and a positive Y value.

• Quadrant 3 is the lower left,

• and it contains all of the points that have both a negative X and a negative Y value.

• And Quadrant 4 is the lower right,

• and it contains all of the points that have a positive X value and a negative Y value.

• Roman Numerals are usually used to label the four quadrants

• and theyre in that order because that’s the order you would encounter the quadrants

• if you started with a line segment from the origin to the coordinate (1,0)

• and then rotated that line counter-clockwise around the origin.

• Doing this sweeps out a shape called a “unit circle

• which is divided into four quadrants just like the coordinate plane.

• Alright, so now you know what the coordinate plane is, and you know how to plot points on it.

• But you might be wondering, “What has this got to do with basic Algebra?”

• Well, Algebra involves many different types of equations and functions that are a lot easier to understand

• if we graph their solutions on the coordinate plane.

• And as you know, the way to really get good at math is to practice what youve learned by doing some exercise problems.

• Thanks for watching Math Antics, and I’ll see ya next time.

• Ohhohhhh

• [snickering]

• [sarcastically] Ahexactly what I wanted.