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.
Quadrant
I x > 0 , or x positive
y > 0 , or y positive
Quadrant
II x < 0 , or x negative
y > 0 , or y positive
Quadrant
III x < 0 , or x negative
y < 0 , or y negative
Quadrant
IV x > 0 , or x positive
y < 0 , or y negative
Each
point in the coordinate system is defined by an ordered pair of the form
(x, y). 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, (x, y) , is called
the coordinates for the point.
The point where the two axes cross is called the origin and
has the coordinates (0, 0).
Note as well that the order of the coordinates is
important. For example, the point (2, 1) is the point that is two units to the right of the origin and then 1 unit up, while
the point (1, 2) 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 (2, – 3) is a point on the graph of
y = (x – 1)2 – 4 while (1, 5) isn’t on the graph. How
do we tell this? All we need to 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.
(2, – 3):
In this case we have x = 2 and y = −2 so
plugging in gives,
So, the coordinates of this point satisfies the equation
and so it is a point on the graph.
(1, 5):
Here we have x =1 and y = 5. 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.
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 of the 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 y = 0 and solve for x. Likewise to find the y – intercepts
for an equation we simply need to set x = 0 and solve for y.
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.
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