Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture, we introduced functions whose graphs in the Cartesian plane
are lines passing through the origin.
These are called "linear functions"
and can be represented by the linear equation "y = mx"
where "m" is a constant whose value can be any real number
positive, negative, or zero.
The slope of a line described by this equation is determined by the value of "m".
Depending upon "m", the slope can be positive
negative
or zero.
However, all graphs described by equations of this form
are lines which pass through the origin.
In this lecture, we will introduce linear functions
whose graphs do not pass through the origin.
These functions are sometimes called "affine" functions.
The equation "y = x" is a simple linear equation
describing a line passing through the origin.
Every point on this line has a y-coordinate
whose value is equal to its x-coordinate.
Now, if instead of equating y to x
we equate y to x plus one
every point's y-coordinate
will be one greater than before.
Every point on this line has a y-coordinate
whose value is one greater
than the value of its x-coordinate.
This equation describes the same line as before
but the entire line is one unit higher
on the Cartesian plane.
Likewise, the equation "y = x + 2"
describes a line two units higher.
Or the equation "y = x - 2"
describes a line two units lower.
In general, adding any real number "b"
to the right side of this equation
will shift the graph up or down by "b" units.
This is true for other linear equations as well.
For example, if we start with the equation "y = 2x"
and add three to the right side of the equation
the graph will be shifted up by three units.
Once again, this is because
the y-coordinate of every point on the graph
is three units greater
than if we had not added three
to the right side of the equation.
For this same reason
taking any linear equation of the form "y = mx"
and adding a constant "b" to the right side
shifts the graph vertically by "b" units.
The graph is shifted up if "b" is positive
or down if "b" is negative.
This shifting of the graph
is referred to as a "translation".
We say that the graph
has been translated vertically by "b" units.
The value of "b" determines the point
at which the graph crosses the y-axis.
This point is called the "y-intercept".
The y intercept has coordinates (0, b).
The first coordinate of the y-intercept
is always zero
because every point on the y-axis
has an x-coordinate of zero.
Since we know that x is zero
we can find the y-coordinate
by making the value of x, zero in the equation.
This makes the first term in the equation
m times 0
which is just zero.
So the first term goes away
leaving us with the equation "y = b".
Therefore when x is zero, y must equal "b".
This tells us that the graph of any equation
of the form "y = mx + b"
always has a y intercept with the coordinates (0,b).
Because the constants "m" and "b" determine the slope and y-intercept
this form of the equation of a line is called the "slope-intercept" form
and any non-vertical line can be described by this equation.
By adjusting the values of the constants "m" and "b"
we can orient the line in any way that we like.
The value of "b" determines the line's vertical position
and the value of "m" determines the line's slope.
Together, these two constants determine the orientation of any non-vertical line
in the Cartesian plane.
In the next lecture, we will show how to calculate the value of a line's slope "m".