SlopeIntercept Form
Linear functions are graphically represented by lines and symbolically written in slopeintercept form as,
y = mx + b,
where m is the slope of the line, and b is the yintercept. We call b the yintercept
because the graph of y = mx + b intersects the yaxis at the point (0, b). We can
verify this by substituting x = 0 into the equation as,
y = m · 0 + b = b.
Notice that we substitute x = 0 to determine where a function intersects the yaxis because
the xcoordinate of a point lying on the yaxis must be zero.
The Definition of Slope :
The constant m expressed in the slopeintercept form of a line, y = mx + b, is
the slope of the line. Slope is defined as the ratio of the rise of the line (i.e. how
much the line rises vertically) to the run of line (i.e. how much the line runs
horizontally).
Definition
For any two distinct points on a line, (x_{1}, y_{1}) and (x_{2}, y_{2}), the slope is,

Intuitively, we can think of the slope as measuring the steepness of a line. The
slope of a line can be positive, negative, zero, or undefined. A horizontal line has
slope zero since it does not rise vertically (i.e. y_{1} − y_{2} = 0), while a vertical line
has undefined slope since it does not run horizontally (i.e. x_{1} − x_{2} = 0).
Zero and Undefined Slope
As stated above, horizontal lines have slope equal to zero. This does not mean
that horizontal lines have no slope. Since m = 0 in the case of horizontal lines,
they are symbolically represented by the equation, y = b. Functions represented by horizontal lines are
often called constant functions. Vertical lines have undefined slope. Since any two
points on a vertical line have the same xcoordinate, slope cannot be computed
as a finite number according to the formula,
because division by zero is an undefined operation. Vertical lines are symbolically
represented by the equation, x = a where a is the xintercept. Vertical lines are
not functions; they do not pass the vertical line test at the point x = a.
Positive Slopes
Lines in slopeintercept form with m > 0 have positive slope. This means for
each unit increase in x, there is a corresponding m unit increase in y (i.e. the
line rises by m units). Lines with positive slope rise to the right on a graph as
shown in the following picture,
Lines with greater slopes rise more steeply. For a one unit increment in x, a line
with slope
m_{1} = 1 rises one unit while a line with slope m_{2} = 2 rises two units
as depicted,
Negative Slopes
Lines in slopeintercept form with m < 0 have negative slope. This means for
each unit increase in x, there is a corresponding m unit decrease in y (i.e. the
line falls by m units). Lines with negative slope fall to the right on a graph as
shown in the following picture,
The steepness of lines with negative slope can also be compared. Specifically,
if two lines have negative slope, the line whose slope has greatest magnitude
(known as the absolute value) falls more steeply. For a one unit increment in x, a line with
slope m_{3} = −1 falls one unit while a line with slope m_{4}= −2 falls two units as
depicted,
Parallel and Perpendicular Lines
Two lines in the xyplane may be classified as parallel or perpendicular based
on their slope. Parallel and perpendicular lines have very special geometric arrangements;
most pairs of lines are neither parallel nor perpendicular. Parallel
lines have the same slope. For example, the lines given by the equations,
y_{1} = −3x + 1,
y_{2} = −3x − 4,
are parallel to one another. These two lines have different yintercepts and will
therefore never intersect one another since they are changing at the same rate
(both lines fall 3 units for each unit increase in x). The graphs of y_{1} and y_{2} are
provided below,
Perpendicular lines have slopes that are negative reciprocals of one another. In
other words, if a line has slope m_{1}, a line that is perpendicular to it will have
slope,
An example of two lines that are perpendicular is given by the following,
These two lines intersect one another and form ninety degree (90°) angles at the
point of intersection. The graphs of y_{3} and y_{4} are provided below,
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In the next
section we will describe how to solve linear equations.
Linear equations
