Basic Linear Functions - Math Antics - Free Educational videos for Students in K-12

Basic Linear Functions - Math Antics - By Mathantics

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00:03	Uh huh . Hi , I'm rob . Welcome to
00:07	Math antics . In this lesson , we're going to
00:09	learn the basics of linear functions , which are really
00:12	common in algebra . We're going to jump right in
00:15	because there's a lot to cover in this video .
00:17	But before we do , if you aren't already familiar
00:20	with topics like graphene and functions , I recommend watching
00:23	our videos about them before continuing on . Okay ,
00:27	so the best way to learn about linear functions is
00:29	to start with one of the most basic linear functions
00:32	of all Y equals X . That's such a simple
00:35	equation that you might be kind of puzzled by it
00:38	at first . But remember the Y variable is simply
00:41	the output of the function and the X variable is
00:44	the input . So all this equation is telling us
00:47	is that the input is exactly the same as the
00:49	output effects is one , then why is also one
00:53	and effects is too then why is also too no
00:56	matter what value you put into the function , you
00:58	get the exact same value out . That might seem
01:01	kind of pointless . But if we grab that function
01:04	on the coordinate plain , you'll see that it forms
01:06	a diagonal line that passes through the origin and splits
01:09	quadrants one and three exactly in half . Notice that
01:13	for any point along the line , the X coordinate
01:16	and the Y coordinate are the same . So Y
01:19	equals X is a very simple linear function . Oh
01:22	, and remember we could use either the variable Y
01:25	or the function notation ffx interchangeably . But we're going
01:29	to use Why in this video to keep it simple
01:32	. Now that we've got that basic case covered ,
01:34	let's look at a slightly more complicated and much more
01:37	versatile linear function Y equals mx . This looks similar
01:42	to the equation Y equals X . But now the
01:44	input variable X is being multiplied by a new variable
01:47	called M . And by choosing different values for em
01:51	, we can make as many different linear functions as
01:53	we want . In fact , if we choose M
01:56	equals one , that would give us Y equals one
01:59	X . Which is just the same function as Y
02:01	equals X . Because multiplying by one doesn't change the
02:04	input value X . But what if we picked a
02:07	different value for M ? Like M equals two .
02:10	That would give us the equation , Y equals two
02:13	X . And if we make a function table for
02:15	that equation and then graph it on the coordinate plain
02:18	, we get a line that looks like this for
02:20	every input value of X . The output Y is
02:23	doubled Despite their differences the lines , Y equals two
02:28	X and Y equals one X . Have something in
02:30	common . They both passed through the origin point of
02:33	the cornet plane 00 Because no matter what value we
02:37	pick for em , if the value X is zero
02:40	, the output , why will also be zero ?
02:42	Since anything multiplied by zero is zero . Okay ,
02:46	what about if we let them equal three instead ,
02:49	that would give us this function table . And this
02:51	line has our graph for every input value of X
02:54	. The output , Y is tripled but it still
02:57	passes through 00 . Do you notice how each time
03:00	we pick a bigger number for M . R .
03:02	Line is getting steeper . Imagine that the line represents
03:05	the site of a mountain or hill that you're climbing
03:08	. Why equals one X would be a steep climb
03:11	. But why equals two . X is steeper and
03:14	Y equals three . X . Is even steeper than
03:16	that in math . The steepness of these lines is
03:20	called their slope . As we choose , bigger and
03:22	bigger values for m the slope of the line increases
03:26	. Did you say slopes ? I love the slopes
03:28	, man . Oh , you should have seen the
03:30	massive air . I just caught off the pipe .
03:32	It was beautiful . I was doing this hard way
03:34	. Front side 1 80 . Oh , man .
03:36	Oh , so awesome . What's that ? All the
03:40	slopes are calling me . I gotta go shred some
03:42	more powder . That sounds pretty impressive . But getting
03:45	back to mathematical slope , if we decided to let
03:49	em equal tin , that would result in a really
03:51	steep line like this . And if m equals 100
03:55	the line slope is so steep that it almost looks
03:57	vertical and it's hard to tell apart from the y
03:59	axis of the coordinate plain . But we can never
04:02	get a truly vertical line with this equation because there's
04:05	no biggest number . The best we can do is
04:08	keep picking bigger and bigger numbers for him and say
04:11	that the slope is approaching infinity as we do that
04:14	and that's fine because a vertical line doesn't qualify as
04:17	a function anyway . So at the y axis we
04:20	seem to have hit a limit . But what if
04:22	we want to make lines that are less steep than
04:24	why equals one X . To do that ? We're
04:26	gonna need to choose some values for em that are
04:28	less than one . Let's start by letting them equal
04:31	1/2 or 0.5 in decimal form . If we make
04:35	a function table for y equals one half X .
04:38	And graph the results , this is what our line
04:40	would look like , yep , that is less steep
04:43	, Let's take it one step further and let them
04:45	equal 1/4 or 0.25 . The function table and graph
04:50	for that equation would look like this . That slope
04:52	is even less as we choose , smaller values for
04:55	M R slope is decreasing and if we keep on
04:58	picking smaller and smaller values for him , like M
05:01	equals 1/10 or M equals 1 100 . You can
05:04	see that our line is looking more and more like
05:06	a completely flat line and it's getting harder to tell
05:09	the difference between it and the horizontal X . Axis
05:13	. Now you might be wondering can we make a
05:15	line that's perfectly horizontal ? Yes . Unlike the case
05:19	when our line was getting steeper and steeper but we
05:21	couldn't ever get it to be a perfectly vertical line
05:24	. We can make our line perfectly horizontal simply by
05:27	choosing M equals zero . Doing that gives us the
05:31	function Y equals zero . Which is just about the
05:34	most boring function you could think of . But it's
05:36	helpful to see because it shows us that a perfectly
05:38	horizontal line has no steepness or a slope of zero
05:43	. It would be just like walking along perfectly flat
05:45	ground . Okay , So when are linear function Y
05:49	equals mx . The variable M . Is the slope
05:52	of the function ? If we start with M equals
05:55	zero and then gradually increase the value of M .
05:58	R line slope gets steeper and steeper it approaches a
06:01	vertical line but it never quite gets there because we
06:04	can't ever really get to infinity . There's always a
06:07	number that's just a little bit bigger . As you
06:10	can see the function Y equals . Mx . Can
06:12	make a lot of different lines but wait , there's
06:15	more . Don't forget about negative numbers . What would
06:18	happen if instead of picking M equals one , we
06:21	pick em equals negative one . If we make a
06:24	function table and graph for that case we end up
06:27	with a line that splits quadrants two and four exactly
06:29	in half it has a slope that's similar in magnitude
06:33	to Y equals one X . But as you move
06:35	from left to right , it's going downhill instead of
06:37	uphill . The slope is negative one basically all of
06:41	the negative values of them give us lines that are
06:44	just mirror images of the lines we get from positive
06:47	values of him . This is M equals one .
06:50	This is M equals negative one . This is M
06:53	equals to this is M equals negative two . This
06:56	is M equals one half . This is M equals
06:59	negative one half . See the pattern . All of
07:02	these possible lines have a positive slope and all of
07:04	these possible lines have a negative slope . And when
07:07	we consider all possible values of M , you can
07:10	see that the equation y equals MX can describe any
07:13	linear function that passes through the origin of the graph
07:17	at 00 . But what if we don't want to
07:19	be limited to lines that pass through the origin of
07:21	the coordinate plain ? No problem . All we have
07:24	to do is add something to this very simple linear
07:27	equation . And I mean literally add something . We're
07:30	just going to add a variable called B to the
07:32	end of our equation , which will give us Y
07:35	equals mx plus B to see what effect this new
07:39	added variable has . Let's set our in value back
07:41	to one and keep it there while we just try
07:44	out different values for B . And we'll also leave
07:47	the graph of Y equals one X . On the
07:49	coordinate plain as a reference to see how it compares
07:52	to our new lines that have the values . Let's
07:55	keep things simple . And start with B equals one
07:58	. That gives us the equation , Y equals one
08:01	X plus one . And if we make a function
08:03	table and graph it , this is the line we
08:05	get notice that it's parallel to the reference line and
08:09	that makes sense because in both equations Y equals one
08:12	, X and Y equals one . X plus one
08:15	. M equals one . So the slope is the
08:17	same for both lines . What's different ? Is that
08:20	the value we chose for ? Be positive one ,
08:22	shifted the entire line up on the coordinate plain by
08:25	one unit . Now the line doesn't pass through zero
08:28	on the y axis , it passes through positive one
08:31	instead . Okay . What will happen if we choose
08:34	B equals positive too ? That gives us the equation
08:38	Y equals one , X plus two . And its
08:41	graph looks like this . It's been shifted up two
08:44	units and now passes through the Y axis at positive
08:46	two . And if we pick B equals three ,
08:49	it would shift the line to intercept the Y axis
08:51	at Y equals three . The bigger the value for
08:54	B , the farther the lion has shifted up .
08:57	But what goes up must come down . Can you
09:00	think of a way to do that to get the
09:02	reference line to shift down instead , yep . Let's
09:05	try using negative numbers for B . Remember adding a
09:08	negative is the same as attracting . If we choose
09:12	B equals negative one , we get y equals one
09:15	. X plus negative one , which is the same
09:17	as Y equals one , X minus one . And
09:20	sure enough that shifts the line down so it crosses
09:23	the y axis at negative one . And if we
09:25	choose the equals negative two it would shift the line
09:28	down so that it crosses the y axis at negative
09:30	two . So do you see what the variable B
09:33	does ? It determines exactly where the line will intercept
09:36	the y axis . It does that because whenever X
09:39	equals zero , which happens only at the y axis
09:43	, the mx term will be zero and we'll be
09:45	left with only be , so when X equals zero
09:49	, why will just equal B . Because of that
09:52	B is called the Y intercept , and as we
09:55	saw earlier , M is called the slope of the
09:58	line . And that's why the equation Y equals mx
10:01	plus B is called the slope intercept form of line
10:05	. The two parameters M . And be determined the
10:08	Lions slope . And it's why intercept . And with
10:11	this simple linear equation , you can describe any possible
10:15	linear function on the coordinate plain . But you might
10:18	be wondering if that's really true . I mean ,
10:20	don't we also need to be able to shift the
10:22	line side to side , nope . And here's why
10:26	, let's say you want to make this line that
10:28	appears to be shifted to the left of the y
10:30	axis . Well if we zoom out just a little
10:32	bit , you'll see that we could get the exact
10:34	same line by shifting our parallel reference line up on
10:38	the y axis . Instead this works because the lines
10:41	are diagonal and they continue on forever in either direction
10:44	. So moving them up and down is equivalent to
10:47	moving them left and right . You can grab any
10:49	two D linear function with just two parameters . The
10:53	multiplied variable M . To rotate the line and the
10:56	added variable B to shift the line . And that's
11:00	why the equation Y equals Mx plus B is so
11:03	important . It's really all you need . But of
11:05	course there's always ways to make things more complicated and
11:09	you'll probably encounter linear equations in a lot of different
11:12	forms . But as long as the equations are truly
11:15	linear functions , you can simplify them into this .
11:17	Y equals mx plus B format . Okay , but
11:22	how do you tell if you have a linear function
11:23	if it's in a different form , will there be
11:26	a linear function equations can only contain first order variables
11:30	? That means that the X and Y terms in
11:32	the equation can't be squared or cubed or raised to
11:35	any powers other than one . So these are all
11:39	examples of linear equations , but these are not .
11:42	And to see how you can rearrange any linear equation
11:45	into the form Y equals mx plus B . Let's
11:49	try to do that to the first equation on this
11:51	list , X minus four equals two times the quantity
11:55	y minus three . We're going to use what we
11:58	learned in previous videos about combining like terms and rearranging
12:02	equations . To get this into the Y equals mx
12:05	plus B form . So we can easily tell what
12:07	the slope and Y intercept would be . Let's see
12:10	we want to get Y all by itself . So
12:12	first we divide both sides by two on the right
12:15	side , the two's cancel . And on the left
12:18	side we have to distribute the division . So we
12:20	get X over two minus 4/2 , which is the
12:24	same as one half , x minus two . The
12:27	next step to get why by itself is to add
12:30	three to both sides on the right , the minus
12:32	three and plus three , cancel . And on the
12:35	left we have minus two plus three which is positive
12:39	one . So the equation becomes one half X plus
12:43	one equals Y or Y equals one half X plus
12:47	one . Now it's in Y equals mx plus B
12:50	form . So we know that the slope is one
12:52	half and the y intercept is positive one . All
12:56	right , so that's the basics of linear equations .
12:59	It's really cool knowing that you can grab any possible
13:02	linear function on the coordinate plain with the simple equation
13:05	Y equals mx plus B . But the most important
13:08	thing you can do to learn about linear functions is
13:11	to practice by doing some exercise problems . That's the
13:14	way to really learn math , as always . Thanks
13:16	for watching Advantix and I'll see you next time learn
13:20	more at math antics dot com .

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