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.
So, if we can graph 
getting the graph of is fairly easy. Let’s take a look at a couple of examples.
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Example 1 Using
transformations sketch the graph of the following functions.
(a) [Solution]
(b) [Solution]
Solution
The first thing to do here is graph the function without
the constant which by this point should be fairly simple for you. Then shift accordingly.
(a)
In this case we first need to graph (the dotted line on the graph below)
and then pick this up and shift it upwards by 3. Coordinate wise this will mean adding 3
onto all the y coordinates of
points on .
Here is the sketch for this one.
[Return to Problems]
(b)
Okay, in this case we’re going to be shifting the graph of
(the dotted line on the graph below) down by
5. Again, from a coordinate standpoint
this means that we subtract 5 from the y
coordinates of points on .
Here is this graph.
[Return to Problems]
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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.
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. They
are exactly opposite than vertical shifts and it’s easy to flip these around
and shift incorrectly if we aren’t being careful.
Vertical and
Horizontal Shifts
Now we can also combine the two shifts we just got done
looking at into a single problem. If we
know the graph of the graph of will be the graph of 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.
Let’s take a look at a couple of examples.
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.
Reflection about the y-axis.
Here is an example of each.
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Example 4 Using
transformation sketch the graph of each of the following.
(a) [Solution]
(b) [Solution]
Solution
(a) Based on the placement
of the minus sign (i.e. it’s
outside the square and NOT inside the square, or ) it
looks like we will be reflecting about the x-axis. So, again, the
means that all we do is change the sign on all the y coordinates.
Here is the sketch of this graph.
[Return to Problems]
(b) Now with this one let’s first address the minus sign under
the square root in more general terms.
We know that we can’t take the square roots of negative numbers,
however the presence of that minus sign doesn’t necessarily cause
problems. We won’t be able to plug
positive values of x into the
function since that would give square roots of negative numbers. However, if x were negative, then the negative of a negative number is
positive and that is okay. For
instance,
So, don’t get all worried about that minus sign.
Now, let’s address the reflection here. Since the minus sign is under the square
root as opposed to in front of it we are doing a reflection about the y-axis. This means that we’ll need to change all
the signs of points on .
Note as well that this syncs up with our discussion on
this minus sign at the start of this part.
Here is the graph for this function.
[Return to Problems]
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