Hi, I’m Rob. Welcome to Math Antics.
In this lesson, we’re gonna learn the basics of Linear Functions, which are really common in Algebra.
We’re gonna jump right in because there’s a lot to cover in this video,
but before we do, if you aren’t already familiar with topics like graphing and functions,
I recommend watching our videos about them before continuing on.
Okay, so the best way to learn about linear functions
is to start with one of the most basic linear function of all: y = x
That’s such a simple equation, that you might be kind of puzzled by it at first.
But remember, the ‘y’ variable is simply the output of the function
and the ‘x’ variable is the input.
So all this equation is telling us is that the input is exactly the same as the output.
If x is 1, then y is also 1,
and if x is 2, then y is also 2.
No matter what value you put into the function, you get the exact same value out.
That might seem kinda pointless,
but if we graph that function on the coordinate plane,
you’ll see that it forms a diagonal line that passes through the origin
and splits quadrants 1 and 3 exactly in half.
Notice that for any point along the line, the x coordinate and the y coordinate are the same.
So y = x is a very simple linear function.
Oh… and remember, we could use either the variable y or the function notation f(x) interchangeably,
but we’re going to use ‘y’ in this video to keep it simple.
Now that we’ve got that basic case covered,
let’s look at a slightly more complicated and much more versatile linear function: y = mx
This looks similar to the equation y = x,
but now the input variable x is being multiplied by a new variable called m.
And by choosing different values for m,
we can make as many different linear functions as we want.
In fact, if we choose m = 1, that would give us y = 1x
which is just the same function as y = x
because multiplying by 1 doesn’t change the input value x.
But what if we picked a different value for m, like m = 2.
That would give us the equation y = 2x
and if we make a function table for that equation and then graph it on the coordinate plane,
we get a line that looks like this.
For every input value of x, the output y is doubled.
Despite their differences, the lines y = 2x and y = 1x have something in common.
They both pass through the origin point of the coordinate plane (0,0)
because no matter what value we pick for m,
if the value x is 0, the output y will also be 0
since anything multiplied by 0 is 0.
Okay, what about if we let m = 3 instead.
That would give us this function table and this line as our graph.
For every input value of x, the output y is tripled, but it still passes through (0,0).
Do you notice how each time we pick a bigger number for m, our line is getting steeper?
Imagine that the line represents the side of a mountain or hill that you’re climbing.
y = 1x would be a steep climb,
but y = 2x is steeper
and y = 3x is even steeper than that!
In math, the steepness of these lines is called their “slope”.
As we choose bigger and bigger values for m, the slope of the line increases.
Did you say slopes? I love the slopes man!
Ah you should have seen the massive air I just caught off the pipe.
It was beautiful!
I was doin’ this hard-way, front-side 180…
oh man… ahhh…so awesome!!!
Oh… what’s that…?
The slopes are callin’ me, I gotta go shred some more powder!
That sounds pretty impressive, but getting back to mathematical slope…
If we decided to let m = 10, that would result in a really steep line like this.
And if m = 100, the line’s slope is so steep that it almost looks vertical
and is hard to tell apart from the the y-axis of the coordinate plane.
But we could never get a truly vertical line with this equation because there’s no biggest number.
The best we can do is keep picking bigger and bigger numbers for m
and say that the slope is “approaching infinity” as we do that.
And that’s fine, because a vertical line doesn’t qualify as a function anyway.
So at the y-axis, we seem to have hit a limit.
But what if we want to make lines that are less steep that y = 1x?
To do that, we’re gonna need to choose some values for m that are less than 1.
Let’s start by letting m equal one-half (or 0.5 in decimal form.)
If we make a function table for y = 1/2 x, and graph the results,
this is what our line would look like.
Yep, that is less steep.
Let’s take it one step further and let m = 1/4 (or 0.25).
The function table and graph for that equation would look like this.
That slope is even less.
As we choose smaller values for m, our slope is decreasing.
And if we keep on picking smaller and smaller values for m,
like m = 1/10
or m = 1/100,
you can see that our line is looking more and more like a completely flat line
and it’s getting harder to tell the difference between it and the horizontal x-axis.
Now you might be wondering, can we make a like that’s perfectly horizontal?
Yes! Unlike the case when our line was getting steeper and steeper
but we couldn’t ever get it to be a perfectly vertical line,
we CAN make our line perfectly horizontal simply by choosing m = 0.
Doing that gives us the function y = 0,
(which is just about the most boring function you could think of).
But it’s helpful to see because it shows us that
a perfectly horizontal line has NO steepness, or a slope of zero.
It would be just like walking along perfectly flat ground.
Okay, so in our linear function y = mx, the variable m is the slope of the function.
If we start with m = 0, and then gradually increase the value of m, our line’s slope gets steeper and steeper.
It approaches a vertical line, but it never quite gets there because we can’t ever really get to infinity…
there’s always a number that’s just a little bit bigger.
As you can see, the function y = mx can make a LOT of different lines.
But wait… there’s more!
Don’t forget about negative numbers.
What would happen if instead of picking m = 1, we pick m = NEGATIVE 1?
If we make a function table and graph for that case,
we end up with a line that splits quadrants 2 and 4 exactly in half.
It has a slope that’s similar in magnitude to y = 1x,
but as you move from left-to-right, it’s going downhill instead of uphill.
The slope is negative 1.
Basically, all of the negative values of m give us lines that are
just mirror images of the lines we get from positive values of m.
This is m = 1, this is m = -1
This is m = 2, this is m = -2
This is m = 1/2, this is m = -1/2
…see the pattern?
All of these possible lines have a positive slope,
and all of these possible lines have a negative slope.
And when we consider ALL possible values of m,
you can see that the equation y = mx can describe ANY linear function that passes through
the origin of the graph at (0,0).
But, what if we don’t want to be limited to lines that pass through the origin of the coordinate plane?
No problem!
All we have to do is add something to this very simple linear equation.
And I mean literally ADD something.
We’re just going to add a variable called ‘b’ to the end of our equation which will give us y = mx + b.
To see what effect this new added variable has,
let’s set our m value back to 1 and keep it there while we just try out different values for b.
And we’ll also leave the graph of y = 1x on the coordinate plane as a reference
to see how it compares to our new lines that have ‘b’ values.
Let’s keep things simple and start with b = 1.
That gives us the equation: y = 1x + 1
And if we make a function table and graph it, this is the line we get.
Notice that it’s parallel to the reference line.
That makes sense because in both equations (y = 1x and y = 1x + 1)
m = 1 so the slope is the same for both lines.
What’s different is that the value we chose for b (positive 1)
shifted the entire line up on the coordinates plane by 1 unit.
Now the line doesn’t pass though zero on the y-axis.
It passes through positive 1 instead.
Okay, what will happen if we choose b = positive 2?
That gives us the equation y = 1x + 2 and its graph looks like this.
It’s been shifted up 2 units and now passes through the y-axis at positive 2.
And if we pick b = 3, it would shift the line to intercept the y-axis at y = 3.
The bigger the value for b, the farther the line is shifted up.
But, what goes up must come down!
Can you think of a way to do that? …to get the reference line to shift DOWN instead?
Yep, let’s try using negative numbers for b.
(Remember, adding a negative is the same as subtracting.)
If we choose b = -1 we get y = 1x + (-1),
which is the same as y = 1x - 1.
And sure enough, that shifts the line down so it crosses the y-axis at -1.
And if we choose b = -2, it would shift the line down so that it crosses the y-axis at -2.
So do you see what the variable b does?
It determines exactly where the line will intercept the y-axis.
It does that because whenever x = 0 (which happens only at the y-axis)
the ‘mx’ term will be zero and we’ll be left with only b.
So when x = 0, y will just equal b.
Because of that, b is the called the “y-intercept”.
And as we saw earlier, m is called the “slope” of the line.
And that’s why the equation y = mx + b is called the “slope-intercept form” of a line.
The two parameters ‘m’ and ‘b’ determine the line’s slope and its y-intercept.
And with this simple linear equation,
you can describe ANY possible linear function on the coordinate plane.
But you might be wondering if that’s really true.
I mean… don’t we also need to be able to shift the line side to side?
Nope, and here’s why.
Let’s say you want to make this line that appears to be shifted to the left of the y-axis.
Well, if we zoom out just a little bit,
you’ll see that we could get the exact same line by
shifting our parallel reference line UP on the y-axis instead.
This works because the lines are diagonal and they continue on forever in either direction,
so moving them up and down is equivalent to moving them left and right.
You can graph any 2D linear function with just two parameter:
the multiplied variable ‘m’ to rotate the line,
and the added variable ‘b’ to shift the line.
And that’s why the equation y = mx + b is so important.
It’s really all you need.
But of course, there’s always ways to make things more complicated
and you’ll probably encounter linear equations in a lot of different forms.
But as long as the equations are truly linear functions,
you can simplify them into this y = mx + b format.
Okay… but how do you tell if you have a linear function if it’s in a different form?
Well, to be a linear function, equations can only contain first order variables.
That means that the x and the y terms in the equation
can’t be squared or cubed or raised to any powers other than 1.
So these are all examples of linear equations,
but these are NOT.
And to see how you can re-arrange any linear equation into the form y = mx + b,
let’s try to do that to the first equation on this list: x - 4 = 2(y - 3)
We’re gonna use what we learned in previous videos about combining like terms
and re-arranging equations to get this into the y = mx + b form
so we can easily tell what the slope and y-intercept would be.
Let’s see… we want to get y all by itself, so first we divide both sides by 2.
On the right side, the twos cancel
and on the left side we have to distribute the division
so we get ‘x over 2’ minus ‘4 over 2’,
which is the same as ‘one-half x’ minus 2.
The next step to get y by itself is to add 3 to both sides.
On the right, the -3 and +3 cancel
and on the left, we have -2 plus 3 which is positive 1,
so the equation becomes 1/2x + 1 = y OR y = 1/2x + 1
Now it’s in y = mx + b form so we know that the slope is one-half,
and the y-intercept is positive one.
Alright, so that’s the basics of linear equations.
It’s really cool knowing that you can graph any possible linear function on the coordinate plane
with the simple equation y = mx + b.
But the most important thing you can do to learn about linear functions is to practice by doing some exercise problems.
That’s the way to really learn math.
As always, thanks for watching Math Antics and I’ll see ya next time.
Learn more at www.mathantics.com