graphical solution Any solution to a pair of simultaneous equations found by plotting the graph of each
equation and visually inspecting the location of a point
of intersection is called a graphical solution. For example, in plotting the graphs of y = 2x + 3 and y = 9 – 4x,
one sees that the lines intersect at the point (1,5) and so
x = 1, y = 5 is a solution to the pair of equations. A
graphical approach is often the only feasible method of
solving a complicated pair of equations. Graphing calculators are equipped with a “trace” button that allows
one to quickly find the coordinates of points of intersection up to some degree of accuracy.
A single equation f(x) = 0 can be solved graphically
by locating the x-intercepts of the function. For example, to find the square root of 2, one can plot a graph
of the function f(x) = x
2 –2 and attempt to identify the
location of the positive x-intercept.
This graphical method makes it clear that a pair of
SIMULTANEOUS LINEAR EQUATIONS:
ax + by = p
cx + dy = q
has either one solution (the lines intersect at a unique
point), no solutions (the lines are parallel), or infinitely
many solutions (the lines coincide).
A DIFFERENTIAL EQUATION of the form
= f(x,y)
can be solved graphically. We seek a curve y = g(x) in
the plane whose slope
at any point (x, y) is given
by f(x,y). Thus if, for a large selection of points (x,y)
across the plane, we draw short line segments of slope
f(x,y), the shapes of curves following these slopes may
be visually apparent. If we are also told the value of the
function y = g(x) at one particular point (x, y), then following the slope of the line segments from that point
onward describes a particular solution to the equation.
See also BISECTION METHOD; NEWTON’S METHOD.
graph of a function A drawing or a visual representation that shows the relationship between two or more
variables is called a graph. It is usual to draw the graph
of a FUNCTION of a single variable y = f(x) on a CARTESIAN COORDINATE system with an x-axis and a y-axis at
right angles. The graph of the function is then the set of
all points (x,y) that satisfy the equation y = f(x) drawn
as a curve in the plane. For example, the set of all
points (x, y) that satisfy y = x 2 forms a PARABOLA,
while the graph of the function y = 3x is a straight line
through the origin with slope 3.
A general approach to graphing a function is to
make a table of (x, y) pairs that satisfy the equation
under consideration and then to locate these points on
a coordinate system. If sufficiently many points are
drawn, then a smooth curve connecting the dots is
likely to be a good representation of the function.
Graphing calculators employ this technique when displaying the graph of a function.
As one develops familiarity with basic equations,
graphing simple formulae becomes a matter of routine.
For example, one can establish that a LINEAR EQUATION of the form y = mx + b yields a straight-line graph
of slope m crossing the y-axis at position b, and that an
equation of the form x
2 + y
2 = r
2 represents a CIRCLE of
radius r. To determine the graphs of more-complicated
functions, scholars—for many decades—could only
resort to the tedious task of making tables and plotting
individual points until a general picture emerged. The
advent of CALCULUS, however, at the turn of the 18th
century brought with it the power to quickly identify
and examine the basic shape and structure of complicated graphs. By examining the first DERIVATIVE of a
function, for instance, one can determine where the
graph increases and decreases, as well as the location of
any local maxima and minima. The second derivative
provides information about the shape of the curve—
whether it is concave up or concave down. This information, together with knowledge of the x- and
y-intercepts of the curve and any ASYMPTOTEs it might
possess, is enough to draw a reasonably accurate picture of the graph without having to plot individual
points. Application of these newly discovered techniques from calculus was literally an eye-opening experience for scholars of the time.
French mathematician NICOLE ORESME (ca.
1323–82) was the first to draw the graph of a function
and to find an interesting interpretation for the area of
the region under it. French lawyer and amateur mathematician PIERRE DE FERMAT (1601–65) developed the
idea further, defining a general procedure for associating curves to formulae. Given a relationship between
two variables A and B, say, Fermat drew a horizontal
reference line for the independent variable A and imagined a second line sliding along this reference line at a
fixed angle whose length B varied according to the
dy
––
dx
dy
––
dx
234 graphical solution
equation and visually inspecting the location of a point
of intersection is called a graphical solution. For example, in plotting the graphs of y = 2x + 3 and y = 9 – 4x,
one sees that the lines intersect at the point (1,5) and so
x = 1, y = 5 is a solution to the pair of equations. A
graphical approach is often the only feasible method of
solving a complicated pair of equations. Graphing calculators are equipped with a “trace” button that allows
one to quickly find the coordinates of points of intersection up to some degree of accuracy.
A single equation f(x) = 0 can be solved graphically
by locating the x-intercepts of the function. For example, to find the square root of 2, one can plot a graph
of the function f(x) = x
2 –2 and attempt to identify the
location of the positive x-intercept.
This graphical method makes it clear that a pair of
SIMULTANEOUS LINEAR EQUATIONS:
ax + by = p
cx + dy = q
has either one solution (the lines intersect at a unique
point), no solutions (the lines are parallel), or infinitely
many solutions (the lines coincide).
A DIFFERENTIAL EQUATION of the form
= f(x,y)
can be solved graphically. We seek a curve y = g(x) in
the plane whose slope
at any point (x, y) is given
by f(x,y). Thus if, for a large selection of points (x,y)
across the plane, we draw short line segments of slope
f(x,y), the shapes of curves following these slopes may
be visually apparent. If we are also told the value of the
function y = g(x) at one particular point (x, y), then following the slope of the line segments from that point
onward describes a particular solution to the equation.
See also BISECTION METHOD; NEWTON’S METHOD.
graph of a function A drawing or a visual representation that shows the relationship between two or more
variables is called a graph. It is usual to draw the graph
of a FUNCTION of a single variable y = f(x) on a CARTESIAN COORDINATE system with an x-axis and a y-axis at
right angles. The graph of the function is then the set of
all points (x,y) that satisfy the equation y = f(x) drawn
as a curve in the plane. For example, the set of all
points (x, y) that satisfy y = x 2 forms a PARABOLA,
while the graph of the function y = 3x is a straight line
through the origin with slope 3.
A general approach to graphing a function is to
make a table of (x, y) pairs that satisfy the equation
under consideration and then to locate these points on
a coordinate system. If sufficiently many points are
drawn, then a smooth curve connecting the dots is
likely to be a good representation of the function.
Graphing calculators employ this technique when displaying the graph of a function.
As one develops familiarity with basic equations,
graphing simple formulae becomes a matter of routine.
For example, one can establish that a LINEAR EQUATION of the form y = mx + b yields a straight-line graph
of slope m crossing the y-axis at position b, and that an
equation of the form x
2 + y
2 = r
2 represents a CIRCLE of
radius r. To determine the graphs of more-complicated
functions, scholars—for many decades—could only
resort to the tedious task of making tables and plotting
individual points until a general picture emerged. The
advent of CALCULUS, however, at the turn of the 18th
century brought with it the power to quickly identify
and examine the basic shape and structure of complicated graphs. By examining the first DERIVATIVE of a
function, for instance, one can determine where the
graph increases and decreases, as well as the location of
any local maxima and minima. The second derivative
provides information about the shape of the curve—
whether it is concave up or concave down. This information, together with knowledge of the x- and
y-intercepts of the curve and any ASYMPTOTEs it might
possess, is enough to draw a reasonably accurate picture of the graph without having to plot individual
points. Application of these newly discovered techniques from calculus was literally an eye-opening experience for scholars of the time.
French mathematician NICOLE ORESME (ca.
1323–82) was the first to draw the graph of a function
and to find an interesting interpretation for the area of
the region under it. French lawyer and amateur mathematician PIERRE DE FERMAT (1601–65) developed the
idea further, defining a general procedure for associating curves to formulae. Given a relationship between
two variables A and B, say, Fermat drew a horizontal
reference line for the independent variable A and imagined a second line sliding along this reference line at a
fixed angle whose length B varied according to the
dy
––
dx
dy
––
dx
234 graphical solution