In
this section we are going to see how knowledge of some fairly simple graphs can
help us graph some more complicated graphs. Collectively the methods we’re
going to be looking at in this section are called transformations.
Vertical
Shifts
The
first transformation we’ll look at is a vertical shift.
Given
the graph of f (x) the graph of g
(x) = f (x) + c will
be the graph of f (x) shifted up by c
units if c is positive and or down by c units if c is
negative.
So,
if we can graph f (x) getting
the graph of g (x) is
fairly easy.
So,
vertical shifts aren’t all that bad if we can graph the “base” function first.
Note as well that if you’re not sure that you believe the graphs in the
previous set of examples all you need to do is plug a couple values of x into
the function and verify that they are in fact the correct graphs.
Horizontal
Shifts
These
are fairly simple as well although there is one bit where we need to be
careful.
Given
the graph of f (x) the graph of g
(x) = f (x + c) will be the
graph of f (x) shifted left by c
units if c is positive and or right by c units if c is
negative.
Now,
we need to be careful here a positive c shifts a graph in the negative
direction and a negative c shifts a graph in the positive direction.
There are exactly opposite than vertical shifts and
it’s easy to flip these around and shift incorrectly if we aren’t being
careful.
Now
we can also combine the two shifts we just got done looking at into a single
problem. If we know the graph of f ( x) the graph of g
(x) = f (x + c) + k will
be the graph of f ( x) shifted left or
right by c units depending on the sign of c and up or down by k
units depending on the sign of k.
Reflections
The
final set of transformations that we’re going to be looking at in this section
aren’t shifts, but instead they are called reflections and there are two of
them.
Reflection
about the x-axis.
Given
the graph of f ( x) then the graph
of g (x) = − f ( x) is the graph of
f (x) reflected about
the x-axis. This means that the signs on the all the y coordinates
are changed to the opposite sign.
Reflection
about the y-axis.
Given
the graph of f ( x) then the graph
of g (x) = f (−x) is the graph of
f (x) reflected about
the y-axis. This means that the signs on the all the x coordinates
are changed to the opposite sign.
Sumber
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