Algebra Basics: Graphing On The Coordinate Plane - Math Antics - Free Educational videos for Students in K-12

Algebra Basics: Graphing On The Coordinate Plane - Math Antics - By Mathantics

Transcript


00:03	Uh huh . Hi , I'm rob . Welcome Math
00:07	antics in this lesson . We're going to learn something
00:09	that's an important foundation for tons of math problems including
00:13	those you'll encounter . We're learning basic algebra . We're
00:17	going to learn about graphing which basically means taking mathematical
00:20	relationships and turning them into pictures . Hey friends ,
00:24	Welcome back to the joy of graphic . We're going
00:26	to pick up right where we left off . We
00:28	already have this nice beautiful function right here . But
00:32	it needs a friend and we're going to do that
00:34	by adding some points . So let's put the next
00:37	point right here . Now all we need to do
00:40	is connect those points because they're all friends and what
00:43	do friends do they stay connected ? Oh and look
00:47	at that , That's beautiful . Well not the kind
00:52	of picture that you'd hang on your wall graphing just
00:55	means making a visual representation of an equation or data
00:59	set so you can understand it better . It's a
01:01	way of helping you literally see how math works when
01:05	math is just a bunch of numbers and symbols on
01:07	a page . It can be pretty abstract and hard
01:09	to relate to . But graphing is like a window
01:12	into the abstract world of math that helps us see
01:15	it more clearly . In fact the focus of our
01:17	lesson today actually looks a bit like a window and
01:20	it's called the coordinate plain . The coordinate plain is
01:23	the platform or stage that are graphing will take place
01:26	on . But to understand how it works , we
01:29	first need to start with its closest relative the number
01:31	line . You remember how a number line works right
01:35	? A number line starts at zero and represents positive
01:37	numbers as you move to the right and negative numbers
01:40	as you move to the left and there's usually marks
01:43	showing where each imager is along the way . Now
01:45	imagine cloning that number line and rotating the copy counterclockwise
01:50	by 90° so that the second number line is perpendicular
01:53	to the first and they intersect at their zero points
01:56	. What we have now is a number of plain
01:58	. It's basically like a two dimensional version of a
02:01	number line . But that second dimension makes it much
02:04	more useful . With a simple one dimensional number line
02:07	, we could show where various numbers were located along
02:10	that line by drawing or plotting points . But no
02:13	matter how many points we plot , they're always on
02:16	the same line . But with the two dimensional number
02:19	plane , we can plot points anywhere in that to
02:22	the area and that opens up a whole new world
02:24	of possibilities . With the one dimensional number , line
02:28	plotting points was easy . You just needed one number
02:30	to tell you where to plot a point . But
02:32	with the two dimensional number plane , you actually need
02:35	to numbers to plot each point . These two numbers
02:38	are called coordinates because they're the same rank or order
02:41	and they work together to specify the location of a
02:44	point on the number plane . In fact , that's
02:47	why the number plane is often referred to as the
02:50	coordinate plain . It's the stage for plotting coordinates ,
02:54	coordinates . Use a special format to help you recognize
02:56	them . The two numbers are put inside parentheses with
02:59	a comma between them as a separator . So when
03:01	you see two comma five or negative seven comma three
03:05	or zero comma 1.5 . You know you're dealing with
03:08	coordinates . Okay to understand how coordinates work , Remember
03:12	that Our number of plane is formed by combining two
03:14	perpendicular number lines from now on , we're going to
03:17	refer to each one of these number lines as an
03:19	axis . One of the axis is horizontal , like
03:22	the horizon , which means the other axis is vertical
03:25	or straight up and down . And they're often called
03:28	the horizontal and vertical axes because of that . But
03:31	even more often , the axes are referred to by
03:33	variable based names . The horizontal axis is called the
03:37	X axis and the vertical axis is called the Y
03:41	axis . Why use variable names ? Well , there's
03:44	two good reasons . The first is that variable names
03:47	are more flexible than horizontal and vertical , which relate
03:50	to specific orientations in space . That may not always
03:54	be relevant . And the second reason is that each
03:57	of the two coordinate numbers is actually a variable that
04:00	relates to a specific position along one of the two
04:03	axes of the coordinate plain . And since those variables
04:06	are usually called X and Y , it makes sense
04:09	to name the two axes . The same way the
04:12	first coordinate number listed will be called X . And
04:14	the second coordinate number listed will be called Why ?
04:17	And we're always going to list the numbers in that
04:18	same order X first and then Why ? So that
04:21	we never get confused about which is which in fact
04:25	coordinates are often called ordered pairs . Because they're a
04:28	pair of numbers that are always listed in the same
04:30	order X . Value first . Why value second ?
04:33	So if you have the coordinates three comma five ,
04:36	that means X equals three , and Y equals five
04:40	. Pretty easy . Right . But now how do
04:42	we actually plot these coordinates or ordered pairs on the
04:46	coordinate plain ? Well , the first number in the
04:48	ordered pair tells you where along the X axis the
04:51	point is located . And the second number in the
04:54	ordered pair tells you where along the Y axis the
04:57	point is located . The two numbers in an ordered
04:59	pair worked together to define a single point , and
05:03	each one of the numbers only gives you half of
05:05	the information about where that point is . To see
05:07	how this works . Let's plot the coordinates 3:02 .
05:11	First we locate the X . Value along the X
05:13	. Axis , which is at three in this case
05:15	. But instead of putting a point there , we
05:17	draw or just imagine a line perpendicular to the X
05:21	axis that goes through the three . We do that
05:24	because the first number in the ordered pair only tells
05:26	us where along the X axis the point is but
05:29	it could be anywhere along the y axis . We
05:31	won't know that until we plot the second number .
05:34	So temporarily we just draw a line there to represent
05:37	every possible point that could have an X . Value
05:39	of three . Next we locate the y value along
05:42	the Y axis , which is at two in this
05:44	case . But again , instead of putting a point
05:47	there , we draw or just imagine a line perpendicular
05:50	to the y axis that goes through the two .
05:53	We do that because the second number in the order
05:55	pair only tells us where along the y axis the
05:57	point is but it could be anywhere along the X
05:59	axis . So we just draw a line there to
06:02	represent every possible point that could have a Y .
06:04	Value of two . Ah But look what we've got
06:07	now , the first line represents all the possible locations
06:11	were X equals three . And the second line represents
06:14	all the possible locations where Y equals two . And
06:17	the exact point where the two lines intersect represents the
06:20	only point in the entire coordinate plane where both X
06:24	equals three and Y equals two . That intersection is
06:28	the location of our point . Pretty cool . Huh
06:32	? And that's a really good way to understand how
06:34	the coordinate plain works . But I want to show
06:36	you an even easier and more intuitive way to actually
06:39	plot points . This more intuitive way involves starting with
06:42	the point at the origin of the coordinate plain and
06:45	then treating the coordinates like a set of simple instructions
06:48	that tell you how far to move your point in
06:50	the X and Y directions , for example , to
06:53	plot the coordinates three comma to like before we start
06:56	by imagining a point at the origin zero comma zero
07:00	. Then we look at the first number in our
07:02	ordered pair to see how far we need to move
07:04	our point in the X direction . Since X is
07:07	positive three , we move our point a distance of
07:10	three units in the positive X . Direction . And
07:13	then from there , since why is positive two ?
07:16	We move our point a distance of two units in
07:18	the positive Y direction . So that's a pretty easy
07:21	method for plotting points . Let's try it a few
07:24	more times . So you get the idea , let's
07:26	plot the coordinates negative four comma three . Again we
07:30	start by imagining a point at the origin and then
07:32	let the coordinates tell us how far to move it
07:34	along the X and Y axes . Since X is
07:38	negative four , we move the point a distance of
07:40	four units . But this time in the negative X
07:42	direction , which is to the left . And then
07:45	since why is positive three , we move the point
07:48	a distance of three units and the positive Y direction
07:51	. Now let's plot the coordinates negative three comma negative
07:54	three . In this case X and Y are both
07:57	negative . So starting with the point at zero comma
08:00	zero , we move it three units in the negative
08:02	X . Direction and then three units in the negative
08:04	Y . Direction . And last let's plot the coordinates
08:08	four common negative 2.5 . Starting at zero comma zero
08:12	, we move the 00.4 units and the positive X
08:15	. Direction . And then 2.5 units in the negative
08:17	Y . Direction . Okay , so we've plotted four
08:21	ordered pairs . The easy way . And did you
08:23	notice that each of these points is located in a
08:25	different region of the coordinate plain ? These four regions
08:28	are called quadrants and their boundaries are defined by the
08:31	two axes of the coordinate plain . The quadrants are
08:34	named one through four , so we can easily refer
08:37	to them in conversations if we need to quadrant ,
08:39	one is the upper right quadrant and it contains all
08:42	of the points where both the X and Y values
08:46	are positive Quadrant two is the upper left and it
08:50	contains all of the points that have a negative x
08:52	. value and a positive y value . Quadrant three
08:56	is the lower left and it contains all of the
08:58	points that have both a negative x . And a
09:00	negative y value . And quadrant four is the lower
09:04	right . And it contains all of the points that
09:06	have a positive x value and a negative y value
09:09	. Roman numerals are usually used to label the four
09:12	quadrants and they're in that order . Because that's order
09:15	, you would encounter the quadrants if you started with
09:17	a line segment from the origin to the coordinate one
09:20	comma zero and then rotated that line counter clockwise around
09:24	the origin . Doing this sweeps out of shape called
09:27	a unit circle , which is divided into four quadrants
09:30	, just like the coordinate plain . All right .
09:33	So now you know what the coordinate plain is and
09:36	you know how to plot points on it . But
09:38	you might be wondering what has this got to do
09:40	with basic algebra ? Well , algebra involves many different
09:44	types of equations and functions that are a lot easier
09:47	to understand if we graph their solutions on the coordinate
09:50	plain , as you know . The way to really
09:53	get good at Math is to practice what you've learned
09:55	by doing some exercise problems . Thanks for watching Math
09:58	Antics and I'll see you next time . Oh ,
10:03	okay . It's ah exactly what I wanted , learn
10:11	more at Math Antics dot com .

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