Slopes and Tangents on a Graph

ok Which is more of a workout, running in downtown Chicago, or running in downtown San Francisco.

now Both cities have traffic and some fantastic waterways to enjoy, but San Francisco has very steep hills.

To get an idea of how steep a hill is, it might make sense to look at how much the elevation changes over a set distance.

This is known as a slope

So let’s say in San Francisco that you’re going over a city block that’s like Mt Everest.

We can define this with something called a slope.

So let’s say that we have a hill where the elevation increases 6 feet over a distance of only 2 feet.

This sounds like mountain goat territory.

now In fact, this means that for each foot that we travel forward, we have to go up 3 feet.

because we have 6 feet to go up / 2 feet forward = 3 feet up per 1 foot forward, 6 /2=3 feet up / per 1 foot forward.

This is our slope, or, how much the elevation changes over a distance of 1, and I’d say it’s very steep.

In Chicago, let’s say that the elevation changes 5 feet over a distance of 50 feet.

now This sounds much more reasonable.

and this means that for every foot you move forward, you only have to go up 0.1 feet, or a little over an inch, because 5 feet up / 50 feet forward = 0.1.

so our slope here is how much the elevation changes over a distance of 0.1.

our slope here in chicago is 0.1.

We might say that in San Francisco if our slope is 3, it’s very steep; it’s mountain goat territory.

In Chicago, our slope is shallow; it’s very small.

but What happens if we have a negative slope

Then we would have an elevation change of -4 feet over a distance of 1 foot.

so before down 1 over

This is a pretty steep drop, i pry helpful on my face.

Mathematically, we can tie all of this back to lines on a graph by simply changing feet into coordinate points.

So let’s look at our steep mountain goat hill.

We’ll call x the distance along the Earth and y the elevation.

so Let’s say the bottom of the hill is at the point (2,1) where x =2 and y =1.

our hill elevation is going to change 6 feet up and 2 feet forward, so our next coordinate is our old coordinate plus 2 feet forward, so 2+2 is 4

our new y coordinate is our old elevation which is 1 + the changing elevation which is 6, so our new y coordinate will be 7

So we’re going from the point (2,1) to the point (4,7).

now In general, you will calculate the slope between two points on a graph like this.

For generality, who will write the coordinates as ( x 1, y 1), like the bottom of the hill is your first point, hints 1, and ( x 2, y 2) are the point of the top of the hill, 2 that’s what you end up.

we calculate the slope, which we’re gonna call m , as delta y / delta x .

Now delta is a mathematician’s way of saying change.

So delta y is the change in y elevation, the change in elevation, delta x is the change in x, in this case the changing distance, or how far forward we’re going to go.

So our slope can be written out as delta y / delta x , which is ( y 2 - y 1) that’s our changing y or delta y / ( x 2 - x 1) that’s our changing x, delta x.

how far forward removing.

if We use this formula to calculate the slope between (2,1) and (4,7).

I’m going to call (2,1) my start point so i’m gonna plug in 2= x 1 start point, and 1= y 1 my start point, and (4,7) is my end point, so 4= x 2 some plug that in here and 7= y 2, so i’m gonna plug that in here.

So my total slope is m = (7 - 1) / (4 - 2) = 6 / 2 = 3, which is exactly we expected from our mouintain go to hill.

Let’s find the slope of the line connecting the point (1,8) and (5,6).

So, again my (1,8) is going to be my start point, and (5,6) is going to be my end point.

my slope is the changing y / changing x.

is the changing y is this distance here, which is just 2.

the changing x is 4.

if i plugging in to this occusion i get -2 / 4 = -0.5, or -1/2.

this negative is because we are line here is pointing downward is not pointing upward.

This is like when I fall down the hill as opposed to the mountain goat traveling gracefully up the hill.

What is the slope of the line given by the equation y = 2 x + 4?

First, let’s graph this by plotting a few points and connecting them with a smooth curve.

so I have the points (0,4), (1,6), (2,8).

each of those pairs satisfiy the equation y = 2x + 4.

so i’m gonna graph them (0,4), (1,6), (2,8) and i’m gonna connect them with the smooth line.

alright so let’s calculate the slope between the first two points, (0,4), (1,6)

The slope m = (6 - 4) that’s my changing y / (1 - 0) that’s my changing x= 2.

alright, great.

what about the slope between these two points, (1,6), (2,8).

my changing y is m = (8 - 6) which is 2/ my changing x is (2 - 1) which is one, so my slope is 2 / 1 = 2.

so The slopes between these two points and these two points is the same

and that makes sense because the graph here is a straight line.

What about the slope of the curve given by the equation y = x ^2?

this is what a graph of this curval look like.

and the Slope here really has no meaning.

do we calculate the slope between the bottom of the curve and this point here, maybe the bottom of a curve end this point over here

maybe we don’t use the bottom of the curve at all, maybe we just calculate this slope

Here the slope has no meaning in particular, so what do we look at instead?

For graphs that are not straight lines, there is no single slope.

so Instead we look at the tangent.

The tangent is the slope of a curve at a single point on that curve.

that is at any point along the curve, there is a line that touches the curve, but does not cross the curve.

so up here, the tangent has a very negative, steep slope.

Here, it’s more shallow.

Here, the slope of the tangent so our straight line just touching our curve, is zero.

Let’s look at the slope of the tangents for another equation.

so here we have got y = x sin( x ).

and let’s imagine I’m walking up this equation.

so here the slope is very high, it is very steep, I’m having a very hard time walking up it.

As I get to the top of the hill, it becomes a little easier, little easier, little easier.

and up of this top, I’m just standing there, the slope of the tangent is zero.

as i keep walking forward and start to go downhill, hopefully i don’t fall downhill and it levels off again, and I can just stand there in the middle if this scrap.

I can walk up back of the hill, stand again at the top of this second hill, before very quickly falling down when the slope gets very steep and negative again.

ok so let’s recap

The slope of a line is a number that describes how steep it is.

A large number corresponds to a steeper slope, like the hills in San Francisco, and a smaller number corresponds to a more shallow slope, like the so-called hills in Chicago.

Positive slopes are pointed upward, and negative slopes are pointed downward.

We calculate the slope by finding the change in y and devided it by the change in x .

that’s like saying the change in elevation, devided by how far forward your walking.

Finally, tangents are slopes at single points along curves.

and we use these when the curve doesn’t have a single slope, but rather the slope is changing at every point.