In this section we need to review some of the basic ideas in
graphing. It is assumed that you’ve seen
some graphing to this point and so we aren’t going to go into great depth
here. We will only be reviewing some of
the basic ideas.
We will start off with the Rectangular or Cartesian
coordinate system. This is just the
standard axis system that we use when sketching our graphs. Here is the Cartesian coordinate system with
a few points plotted.
The horizontal and vertical axes, typically called the x-axis
and the y-axis respectively, divide the coordinate system up into
quadrants as shown above. In each
quadrant we have the following signs for x
and y.
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Quadrant I
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 ,
or x positive
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,
or y positive
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Quadrant II
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,
or x negative
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,
or y positive
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Quadrant III
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,
or x negative
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,
or y negative
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Quadrant IV
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,
or x positive
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,
or y negative
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Each point in the coordinate system is defined by an ordered pair of the form . The first number listed is the x-coordinate
of the point and the second number listed is the y-coordinate of the
point. The ordered pair for any given
point, ,
is called the coordinates for the
point.
The point where the two axes cross is called the origin and has the coordinates .
Note as well that the order of the coordinates is
important. For example, the point is the point that is two units to the right of
the origin and then 1 unit up, while the point is the point that is 1 unit to the right of
the origin and then 2 units up.
We now need to discuss graphing an equation. The first question that we should ask is what
exactly is a graph of an equation? A
graph is the set of all the ordered pairs whose coordinates satisfy the equation.
For instance, the point is a point on the graph of while isn’t on the graph. How do we tell this? All we need to do is take the coordinates of the
point and plug them into the equation to see if they satisfy the equation. Let’s do that for both to verify the claims
made above.
:
In this case we have and
so plugging in gives,
So, the coordinates of this point satisfies the equation and
so it is a point on the graph.
:
Here we have and . Plugging these in gives,
The coordinates of this point do NOT satisfy the equation
and so this point isn’t on the graph.
Now, how do we sketch the graph of an equation? Of course, the answer to this depends on just
how much you know about the equation to start off with. For instance, if you know that the equation
is a line or a circle we’ve got simple ways to determine the graph in these
cases. There are also many other kinds
of equations that we can usually get the graph from the equation without a lot
of work. We will see many of these in
the next chapter.
However, let’s suppose that we don’t know ahead of time just
what the equation is or any of the ways to quickly sketch the graph. In these cases we will need to recall that
the graph is simply all the points that satisfy the equation. So, all we can do is plot points. We will pick values of x, compute y from the
equation and then plot the ordered pair given by these two values.
How, do we determine which values of x to choose? Unfortunately,
the answer there is we guess. We pick
some values and see what we get for a graph.
If it looks like we’ve got a pretty good sketch we stop. If not we pick some more. Knowing the values of x to choose is really something that we can only get with
experience and some knowledge of what the graph of the equation will probably look like. Hopefully, by the end of this course you will
have gained some of this knowledge.
Let’s take a quick look at a graph.
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Example 1 Sketch
the graph of .
Solution
Now, this is a parabola and after the next chapter you
will be able to quickly graph this without much effort. However, we haven’t gotten that far yet and
so we will need to choose some values of x,
plug them in and compute the y
values.
As mentioned earlier, it helps to have an idea of what
this graph is liable to look like when picking values of x. So, don’t worry at this
point why we chose the values that we did.
After the next chapter you would also be able to choose these values
of x.
Here is a table of values for this equation.
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x
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y
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-2
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5
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-1
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0
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0
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-3
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1
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-4
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2
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-3
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3
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0
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4
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5
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Let’s verify the first one and we’ll leave the rest to you
to verify. For the first one we simply
plug into the equation and compute y.
Here is the graph of this equation.
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Notice that when we set up the axis system in this example,
we only set up as much as we needed. For
example, since we didn’t go past -2 with our computations we didn’t go much
past that with our axis system.
Also, notice that we used a different scale on each of the
axes. With the horizontal axis we
incremented by 1’s while on the vertical axis we incremented by 2. This will often be done in order to make the
sketching easier.
The final topic that we want to discuss in this section is
that of intercepts. Notice that the graph in the above example
crosses the x-axis in two places and
the y-axis in one place. All three of these points are called
intercepts. We can, and often will be,
more specific however.
We often will want to know if an intercept crosses the x or y-axis
specifically. So, if an intercept
crosses the x-axis we will call it an
x-intercept. Likewise, if an intercept crosses the y-axis we will call it a y-intercept.
Now, since the x-intercept
crosses x-axis then the y coordinates of the x-intercept(s) will be zero. Also, the x
coordinate of the y-intercept will be
zero since these points cross the y-axis. These facts give us a way to determine the
intercepts for an equation. To find the x-intercepts for an equation all that we
need to do is set and solve for x. Likewise to find the y-intercepts for an equation we simply
need to set and solve for y.
Let’s take a quick look at an example.
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Example 2 Determine
the x-intercepts and y-intercepts for each of the following
equations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
As verification for each of these we will also sketch the
graph of each function. We will leave
the details of the sketching to you to verify. Also, these are all parabolas and as mentioned
earlier we will be looking at these in detail in the next chapter.
(a)
Let’s first find the y-intercept(s). Again, we do this by setting and solving for y. This is usually the
easier of the two. So, let’s find the y-intercept(s).
So, there is a single y-intercept
: .
The work for the x-intercept(s)
is almost identical except in this case we set and solve for x. Here is that work.
For this equation there are two x-intercepts : and . Oh, and you do remember how to solve quadratic equations right?
For verification purposes here is sketch of the graph for
this equation.
[Return to Problems]
(b)
First, the y-intercepts.
So, we’ve got a single y-intercepts. Now, the x-intercept(s).
Okay, we got complex solutions from this equation. What this means is that we will not have
any x-intercepts. Note that it is perfectly acceptable for
this to happen so don’t worry about it when it does happen.
Here is the graph for this equation.
Sure enough, it doesn’t cross the x-axis.
[Return to Problems]
(c)
Here is the y-intercept
work for this equation.
Now the x-intercept
work.
In this case we have a single x-intercept.
Here is a sketch of the graph for this equation.
Now, notice that in this case the graph doesn’t actually
cross the x-axis at . This point is still called an x-intercept however.
[Return to Problems]
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We should make one final comment before leaving this
section. In the previous set of examples
all the equations were quadratic equations.
This was done only because the exhibited the range of behaviors that we
were looking for and we would be able to do the work as well. You should not walk away from this discussion
of intercepts with the idea that they will only occur for quadratic
equations. They can, and do, occur for
many different equations.