Now we need to discuss graphing functions. If we recall from the previous section we
said that 
is nothing more than a fancy way of writing y.
This means that we already know how to graph functions. We graph functions in exactly the same way
that we graph equations. If we know ahead
of time what the function is a graph of we can use that information to help us
with the graph and if we don’t know what the function is ahead of time then all
we need to do is plug in some x’s
compute the value of the function (which is really a y value) and then plot the points.
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Example 1 Sketch
the graph of .
Solution
Now, as we talked about when we first looked at graphing
earlier in this chapter we’ll need to pick values of x to plug in and knowing the values to pick really only comes
with experience. Therefore, don’t
worry so much about the values of x
that we’re using here. By the end of this
chapter you’ll also be able to correctly pick these values.
Here are the function evaluations.
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x
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f(x)
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-1
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-7
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0
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0
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1
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1
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2
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2
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3
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9
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Here is the sketch of the graph.
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So, graphing functions is pretty much the same as graphing
equations.
There is one function that we’ve seen to this point that we
didn’t really see anything like when we were graphing equations in the first
part of this chapter. That is piecewise
functions. So, we should graph a couple
of these to make sure that we can graph them as well.
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Example 2 Sketch
the graph of the following piecewise function.
Solution
Okay, now when we are graphing piecewise functions we are
really graphing several functions at once, except we are only going to graph
them on very specific intervals. In
this case we will be graphing the following two functions,
We’ll need to be a little careful with what is going on
right at since technically that will only be valid
for the bottom function. However,
we’ll deal with that at the very end when we actually do the graph. For now, we will use in both functions.
The first thing to do here is to get a table of values for
each function on the specified range and again we will use in both even though technically it only
should be used with the bottom function.
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x
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-2
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0
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-1
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3
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0
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4
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1
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3
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x
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1
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1
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2
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3
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3
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5
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Here is a sketch of the graph and notice how we denoted
the points at . For the top function we used an open dot
for the point at and for the bottom function we used a closed
dot at . In this way we make it clear on the graph
that only the bottom function really has a point at .
Notice that since the two graphs didn’t meet at we left a blank space in the graph. Do NOT connect these two points with a
line. There really does need to be a
break there to signify that the two portions do not meet at .
Sometimes the two portions will meet at these points and
at other times they won’t. We
shouldn’t ever expect them to meet or not to meet until we’ve actually
sketched the graph.
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Let’s take a look at another example of a piecewise
function.
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Example 3 Sketch
the graph of the following piecewise function.
Solution
In this case we will be graphing three functions on the
ranges given above. So, as with the
previous example we will get function values for each function in its
specified range and we will include the endpoints of each range in each
computation. When we graph we will
acknowledge which function the endpoint actually belongs with by using a closed
dot as we did previously. Also, the
top and bottom functions are lines and so we don’t really need more than two
points for these two. We’ll get a
couple more points for the middle function.
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x
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-3
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0
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-2
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1
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x
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-2
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4
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-1
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1
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0
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0
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1
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1
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x
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1
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1
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2
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0
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Here is the sketch of the graph.
Note that in this case two of the portions met at the
breaking point and at the other breaking point, ,
they didn’t meet up. As noted in the
previous example sometimes they meet up and sometimes they won’t.
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