Professor Dave again, let’s talk about slopes and intercepts.
In math, when we use the word slope, it doesn’t mean we are going to hit the slopes, although
there is some similarity between the downward slope of a mountain and the meaning of this
word in math.
It means a line’s rate of change in the vertical direction, which can be shallow or
steep, just like a mountain.
To understand how the slope of a line relates to the equation of the line, let’s introduce
a common form for linear equations, Y equals MX plus B.
We already know about X and Y, these are the independent variable and dependent variable
respectively.
But these other two terms represent characteristics of the line.
M represents the slope of the line, which is as we said, essentially how steep the line
is, and B represents the Y-intercept, which is the Y coordinate for the point where the
line crosses the Y axis.
Let’s start by looking at the simplest example possible, Y equals X.
We already learned how to plot this line, so here it is, with every point showing an
ordered pair where the X and Y values are equivalent.
To calculate the slope, we must understand that this is equal to rise over run, which
means change in Y over change in X.
We can pick any two points on this line, and with these points we take the difference in
their Y values, and divide by the difference in their X values.
Y is up and down, or the rise, and X is side to side, or the run.
Rise over run.
Let’s pick these two points, two two and four four.
Rise over run can be represented this way, Y two minus Y one over X two minus X one,
so we just plug the coordinates in.
Y two is four, Y one is two, X two is four, and X one is two.
We simplify and get two over two, or one.
The slope of this line is one.
That should make sense, because Y equals X is in Y equals MX plus B form, it’s just
that M is one and B is zero.
One X is just X, and B is zero, because the line crosses the Y axis when Y equals zero,
so we get this very simple equation, Y equals X.
If a line has a slope that is greater than one, it will tilt upwards like this, steeper
and steeper, climbing towards infinity, until it reaches complete verticality, at which
point the slope is undefined, because the line rises all the way to infinity in the
Y direction without any run at all, and infinity over zero is undefined.
If instead the slope is less than one, the line tilts downwards this way, the slope getting
smaller and smaller, until it is completely horizontal, at which point the slope will
become zero.
The line runs all the way to infinity in the X direction without rising at all, and zero
over infinity is zero.
So vertical lines have an undefined slope, horizontal lines have zero slope, and everything
in between has some slope between zero and infinity.
If we continue past the horizontal, the slope becomes increasingly negative, because as
we run in the positive direction, the rise is negative, or moving downwards.
The slope will increase in the negative direction, approaching negative infinity until vertical,
where the slope becomes undefined.
Let’s just quickly mention that any two points define a line.
You could draw any two points at random on the coordinate plane, and there is a line
that connects them.
However, all lines contain an infinite number of points, since the coordinates can become
infinitely precise, and any point on the line represents an X and Y value that qualifies
as a solution to the equation.
We could pick two points, like three two and five seven, and draw a line between them.
We could also calculate the slope.
Seven minus two is five, and five minus three is two, so the slope is five halves.
Always remember that when calculating slope, it doesn’t matter which point is point one
and which is point two, but the coordinates of each point must remain together, so whichever
Y value is Y two, the X value in that point must be X two, and not X one.
That would lead to an incorrect slope calculation.
Also, one point and a slope can define a line.
We could say that a line passes through two four and has a slope of three halves.
To draw this line, we would just go to the the point two four, and start applying the
slope.
Up three, over two.
This could also be down three, two the left, because negative three over negative two also
equals three halves, and that’s how the slope, which we can regard as the rate of
change, makes sense regardless of which direction you travel.
Now that we understand slopes and Y intercepts, let’s check comprehension.