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## Get ready for AP® Calculus

### Course: Get ready for AP® Calculus > Unit 2

Lesson 3: Point-slope form# Intro to point-slope form

CCSS.Math: , , ,

Point-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). Watch this video to learn more about it and see some examples. Created by Sal Khan.

## Want to join the conversation?

- Why is slope referred by 'm'? Is it some kind of short form?(43 votes)
- The truth of it is, no-one really knows. There are several stories around that say it's the first letter for slope in various languages, or that it's derived from the Latin mons (which means mountain), but none of these has any evidence to substantiate them.(82 votes)

- Wait then what form is y = mx + b(17 votes)
- just a reorganized version of point-slope.. they say the same thing, just with different parts.(15 votes)

- At about00:58seconds or so. He says that those triangles are the deltas. I understand that but for full formula for slope does it matter which y or x goes first? Would you still get the same answer?(8 votes)
- Slope is always rise over run. It doesn't matter which one you find first, but make sure they're in the proper place.

Consider a line with rise 5 and run 4. The rise/run way is 5/4. But the run/rise way is 4/5. That is a different value, and would give us with a completely different line.(22 votes)

- I am a student teacher and I have difficulty in thinking about an activity that will lead to this subject. They want it to be a discovery activity that will also serve as a motivational activity for this lesson. I refered to books, but there's no discovery activity for this lesson. Do you have anything in mind?(6 votes)
- Hi, Paula. Here are some ideas:

1. One way to think about point-slope form is as a**rearrangement**of the*slope formula*.

If you ask your kids to manipulate`m = (y - k)/(x - h)`

, perhaps one will come up with`(y - k) = m(x - h)`

.

2. Another way to think about point-slope form is as a**transformation**of the*canonical line*`y = mx`

: That is to say,`(y - k) = m(x - h)`

is the end result of a vertical translation by k units, and a horizontal translation by h units, performed in either order.

3. Also food for thought: Given a point`(h,k)`

and a slope`m`

, the equation`(y - k) = m(x - h)`

is guaranteed to evaluate as`0 = m·0 = 0`

.

Of course 0 is the product of any number and 0. This conceptually echoes how polynomial factors yield roots, based on the fact that any zero product must have one or more zero factors (aka the**Zero Product Property**).(10 votes)

- what is the traditional point-slope formula?(7 votes)
- My math teacher uses an equation of y-y1 = m(x-x1). Is this equation equal to the one in the video? If so, what would the (a,b) be taking the place of?(6 votes)
- Yes they are the same, and (a,b) would take the place of (x1,y1)(9 votes)

- I don't understand pointform at all please explain someone(5 votes)
- The point-slope form is very useful when you don't have your y-intercept. It is used to write equations when you only have your slope and a point.

Point-slope form: y-a = m(x-b).

For example, your slope (m) is 3 and your point (a,b) is 9,10.

You would substitute your y-coordinate for a, and your x- coordinate for b. Your new equation would look like this: y-10 = 3(x-9). If you simplify this, then you will get your basic slope-intercept form: y=mx+b! I hope this made sense!(6 votes)

- I want to join in the discussions but I dont have any questions and I’m not very good at explaining things.(5 votes)
- what would i replace M with(4 votes)
- m in here is the slope or gradient. The way I find easier to think about it is how much y increases or decreases per change of one position in x.

Hope this wasn't too confusing!(3 votes)

- what do you need for the equation to work??(4 votes)
- You just need to plug in all the information you know, to find what you dont know

Hope that helps :)(2 votes)

## Video transcript

So what I've drawn here
in yellow is a line. And let's say we know two
things about this line. We know that it
has a slope of m, and we know that the point
a, b is on this line. And so the question that we're
going to try to answer is, can we easily come
up with an equation for this line using
this information? Well, let's try it out. So any point on this line,
or any x, y on this line, would have to
satisfy the condition that the slope
between that point-- so let's say that this
is some point x, y. It's an arbitrary
point on the line-- the fact that it's
on the line tells us that the slope between a, b
and x, y must be equal to m. So let's use that knowledge to
actually construct an equation. So what is the slope
between a, b and x, y? Well, our change in y--
remember slope is just change in y over change in x. Let me write that. Slope is equal to change
in y over change in x. This little triangle character,
that's the Greek letter Delta, shorthand for change in. Our change in y--
well let's see. If we start at y is equal
to b, and if we end up at y equals this arbitrary
y right over here, this change in y right over
here is going to be y minus b. Let me write it in
those same colors. So this is going to be y
minus my little orange b. And that's going to be
over our change in x. And the exact same logic--
we start at x equals a. We finish at x
equals this arbitrary x, whatever x we
happen to be at. So that change in x is
going to be that ending point minus our starting
point-- minus a. And we know this is the slope
between these two points. That's the slope between
any two points on this line. And that's going
to be equal to m. So this is going
to be equal to m. And so what we've
already done here is actually create an equation
that describes this line. It might not be in any form
that you're used to seeing, but this is an
equation that describes any x, y that satisfies this
equation right over here will be on the line because
any x, y that satisfies this, the slope between that x, y
and this point right over here, between the point a, b,
is going to be equal to m. So let's actually now
convert this into forms that we might
recognize more easily. So let me paste that. So to simplify this expression
a little bit, or at least to get rid of the x minus
a in the denominator, let's multiply both
sides by x minus a. So if we multiply both sides
by x minus a-- so x minus a on the left-hand side and
x minus a on the right. Let me put some
parentheses around it. So we're going to multiply
both sides by x minus a. The whole point of that is you
have x minus a divided by x minus a, which is just
going to be equal to 1. And then on the
right-hand side, you just have m times x minus a. So this whole thing
has simplified to y minus b is equal
to m times x minus a. And right here, this is
a form that people, that mathematicians, have
categorized as point-slope form. So this right over here
is the point-slope form of the equation that
describes this line. Now, why is it called
point-slope form? Well, it's very easy to
inspect this and say, OK. Well look, this is the
slope of the line in green. That's the slope of the line. And I can put the two points in. If the point a, b
is on this line, I'll have the slope times x
minus a is equal to y minus b. Now, let's see
why this is useful or why people like to
use this type of thing. Let's not use just a, b
and a slope of m anymore. Let's make this a little
bit more concrete. Let's say that someone tells you
that I'm dealing with some line where the slope is equal
to 2, and let's say it goes through the
point negative 7, 5. So very quickly, you
could use this information and your knowledge
of point-slope form to write this in this form. You would just say,
well, an equation that contains this point and has this
slope would be y minus b, which is 5-- y minus the
y-coordinate of the point that this line contains--
is equal to my slope times x minus the x-coordinate
that this line contains. So x minus negative 7. And just like that, we have
written an equation that has a slope of 2
and that contains this point right over here. And if we don't like the x minus
negative 7 right over here, we could obviously
rewrite that as x plus 7. But this is kind of the
purest point-slope form. If you want to simplify
it a little bit, you could write it as y minus
5 is equal to 2 times x plus 7. And if you want to see that this
is just one way of expressing the equation of this line--
there are many others, and the one that we're
most familiar with is y-intercept form--
this can easily be converted to
y-intercept form. To do that, we just have
to distribute this 2. So we get y minus 5 is equal
to 2 times x plus 2 times 7, so that's equal to 14. And then we can get rid of
this negative 5 on the left by adding 5 to both
sides of this equation. And then we are left with,
on the left-hand side, y and, on the right-hand
side, 2x plus 19. So this right over here
is slope-intercept form. You have your slope
and your y-intercept. So this is slope-intercept form. And this right up here
is point-slope form.