How is the slope measureand y-interceptof a line determined?
On a graph, the slope is the line in a plane completely specified by a point and a num-
ber. Mathematicians know that when an equation is written as ymxb,they can
determine the slope measure and y-intercept (or slope intercept) of the line. In this
case, the m(or xcoefficient) is called the slope measureand the y-intercept are the
coordinates that solve the equation.
For example, for y 2 x4, the slope measure is 2 and y-intercept is the
coordinates (0, 4). To test this result, replace the xwith the first coordinate (0); 2
0 0; therefore, 4 is left, resulting in y(or the y-intercept) being 4.What does point-slope equation of a linerefer to in analytic geometry?
A point-slope equation of a line refers to Cartesian coordinates: If a point has coordi-
nates (x 1 , y 1 ) on a line and a slope equal to m,then the equation yy 1 m(xx 1 )
is the point-slope equation. Point-slope is most often used when finding an equation
of a line.Why are functionsimportant to analytic geometry?
The intricacies of functions are important mainly because the function (and the graph
of all the points in a function) is one of the basic foundations of analytic geometry.
This becomes even more evident when trying to solve complex equations or equations
with more than one variable.
For example, when determining the solution to an equation, such as 3x 4 y8,
the two resulting numbers—called a set of ordered pairs of numbers—is called a rela-
tion. In turn, a function then becomes a relation in which each first element, such as
x,is matched exactly with a second element, such as y. In other words, a function can
take on a definite value (or values) when certain values are assigned to other quanti-
ties or independent variables of the function. (For more information about functions,
196 see “Foundations of Mathematics” and “Algebra.”)
How are graphs used to solve equations?
A
ny figure represented by an equation in two variables is a curve in two
dimensions. For example, the equation 3x 4 y8 is also called a linear
functionbecause the solution represents a line (which is considered a “curve” in
geometry) on a graph. There are also graphs of equations in three variables, with
the solution as a surface—or the two-dimensional “outer layer” of a three-
dimensional object.