1.7 Graphing Shortcuts
The
graph of an equation in x and y is the set of all points with
coordinates (x,y) satisfying the equation.
In the case of a function y = f ( x
), the graph of f is the point set {
(x,y) | y = f ( x
) }.
The
most fundamental way to sketch the graph of a function is to
(1) Select an interval on the x-axis encompassing all or part of the
domain of the function
(2) Select a finite set of points in the domain
of the function lying in that selected interval
(3) For each of the selected values of x, the corresponding value of y is computed using the formula for
y = f ( x ).
(4)
Plot this finite set of points in the x
y plane and sketch the graph by
connecting the points, being careful not to connect two adjacent points unless
all the input values lying between the two points are in the domain of the
function.
However
there are certain shortcuts which may make the task of sketching graphs easier
by reducing the number of points which must be plotted. These shortcuts may be denoted as
(1)
symmetry
(2)
shifting, i.e. translation
(3)
stretching or shrinking
(4)
reflecting
These
shortcuts apply not only to the graphs of functions (vertical line test) but to
all graphs having an x y equation.
Symmetry
A
graph is symmetric with respect to the y-axis
if the point ( - x, y ) is on the graph whenever the point
( x, y
) is on the graph. This means that if
one replaces every occurrence of x in
the equation with ( - x ) [be sure to
put parentheses around the - x] and
then simplifies the equation, the resulting equation will be equivalent to the
original equation.
Exercise 1.7.1
Let
on the interval [ -
3, 3 ]. Replace the x in the equation with ( - x ) and simplify. Did you get the same equation? Sketch the graph and notice that for every
point ( x, y ) on the graph, the mirror image of that point in the y axis, namely the point ( - x, y
), is also on the graph.
A function
which is symmetric with respect to the y-axis
is called an even function and
has the special property that f ( - x ) = f ( x ) for every x in the domain of the function.
Exercise 1.7.2
Show
that the function 
is an even function.
A
graph is symmetric with respect to the x-axis
if the point ( x, - y ) is on the graph whenever the point
( x, y
) is on the graph. This means that if
one replaces every occurrence of y in
the equation with ( - y ) [be sure to
put parentheses around the - y] and
then simplifies the equation, the resulting equation will be equivalent to the
original equation. Note that if y is not 0 then ( x, y ) and ( x, - y
) are two points which lie on the same vertical line, so the graph would not be
the graph of a function in that case. [Why not?]
Exercise 1.7.3
Let
for y in the interval [ - 2, 2 ]. Replace the y in the equation with ( - y
) and simplify. Did you get the same
equation? Sketch the graph and notice
that for every point ( x, y ) on the graph, the mirror image of
that point in the x axis, namely the
point ( x, - y ), is also on the graph.
A
graph is symmetric with respect to the origin if the point ( - x,
- y ) is on the graph whenever
the point
( x, y
) is on the graph. This means that if
one replaces every occurance of x in
the equation with ( - x ) and every
occurance of y in the equation with (
- y ) [be sure to put parentheses
around the - x and - y]
and then simplifies the equation, the resulting equation will be equivalent to
the original equation. Notice that the
point ( - x, - y
) is the same distance from the origin
as the point ( x, y ) , except it is on the opposite side
of the origin.
Exercise 1.7.4
Let
for x in the interval [ - 2, 2 ]. Replace
the x in the equation with ( - x ) and replace the y in the equation with ( - y
) and simplify. Did you get the same
equation? Sketch the graph and notice
that for every point ( x, y ) on the graph, the point ( - x, - y
), is also on the graph.
A
function which is symmetric with respect to the origin is called an odd function and has the special property that f ( -
x ) = - f ( x ) for every x in the
domain of the function.
Exercise 1.7.5
Show
that the function 
is an odd function.
By noticing
the symmetry of a graph, one need plot only half as many points, obtaining the
other points of the graph by symmetry.
Translation
The
horizontal translation principle states that replacing each occurrence
of x in an x y equation with
( x
- h ) [be sure to put parentheses
around the x - h ] produces an equation whose graph lies h units horizontally from the graph of the original equation. The h
can be any real number.
Exercise 1.7.6
Sketch
the graph of 
, 
and 
. Notice the
application of the horizontal translation principle
Stated
in terms of function notation, the
horizontal translation principle states that the graph of
y = f ( x - h ) is shifted h units horizontally from the graph of y = f ( x ).
Thus the graph of y = f ( x
- 2 ) will lie two units to the right of the graph of y = f ( x ) and the graph of y = f
( x + 3 ) = f ( x - ( - 3 ) ) will
lie three units to the left of y = f ( x
).
Exercise 1.7.7
Sketch
the graph of 
. Then sketch the
graphs of 
and 
.
The
vertical translation principle states that replacing each occurrence of y in an x y equation with ( y - k ) [be sure to put parentheses around
the y - k ] produces an equation whose graph lies k units vertically from the graph of the original equation. The k
can be any real number.
Exercise 1.7.8
Sketch
the graph of 
, 
and 
. Notice the
application of the vertical translation principle
Stated
in terms of function notation, the
horizontal translation principle states that the graph of
y - k = f ( x ) [alternately, y = f ( x ) + k ] is shifted
k units vertically from the graph of y = f
( x ). Thus the graph of y - 2
= f ( x ) will lie two units
above the graph of y = f ( x
) and the graph of y + 3= f ( x ) will lie three units below y = f
( x ).
Exercise 1.7.9
Sketch
the graph of 
.Then sketch the graphs of 
and 
.
Both
the horizontal and vertical translation principles may be applied to the same
problem.
Exercise 1.7.10
Sketch
the graph of 
using both the
horizontal and vertical translation principles.
Stretching/shrinking
The
horizontal stretching/shrinking principle states that replacing each
occurrence of x in an x y equation with x
/ c stretches the graph horizontally
by a factor of c when c > 1 and shrinks the graph
horizontally by a factor c when 0
< c < 1.
Exercise 1.7.11
Sketch
the graph of 
together with the
graph of
. Notice that the
first graph is twice as wide as the second.
Exercise 1.7.12
Sketch
the graph of 
together with the
graph of 
. Notice that the
first graph is only one-half as wide as the second.
The
vertical stretching/shrinking principle states that replacing each
occurrence of y in an x y equation with y
/ c stretches the graph vertically by
a factor of c when c > 1 and shrinks the graph
vertically by a factor c when 0 < c < 1.
Exercise 1.7.13
Sketch
the graph of 
together with the
graph of 
. Notice that the
first graph is twice as tall as the second.
Exercise 1.7.14
Sketch
the graph of 
together with the
graph of 
. Hint: Compare this problem to exercise 1.7.12.
Exercise 1.7.15
Sketch
the graph of 
[that is, 
], together with the graph of 
. Notice that the first
graph is only one-half as tall as the second.
Exercise 1.7.16
Sketch
the graph of 
together with the
graph of 
. Hint: Compare to exercise 1.7.15.
Reflection
The
vertical reflection principle states that replacing each occurrence of x in an x y equation with ( - x
)[be sure to put parentheses around the
- x ] reflects the graph of
the original equation about the y-axis.
Exercise 1.7.17
Sketch
the graphs of 
and 
.
The
horizontal reflection principle
states that replacing each occurrence of y
in an x y equation with
( - y
)[be sure to put parentheses around the - y
] reflects the graph of the original equation about the x-axis.
Exercise 1.7.18
Sketch
the graphs of 
and 
, i.e. 
.
Exercise 1.7.19
Sketch
the graph of 
using the principles
of reflection and translation.