Second degree equations of the form

occur frequently and its graph is called a parabola.

In fact, there are other second degree equations called ellipses and hyperbolas. But before o deep study about them, a perfect understanding of parabolas is strictly required. Because, parabolas are the key elements in understanding of analytic plane and analyses of whose behaviors provide us the first course in understanding the second degree equations.

Therefore, it is important for teachers to be sure whether their students have enough ideas about "what is going on with parabolas".

It is obvious that, drowing and understanding the graphs of the equation could be more or less complex with respect to the values of the coeefficients a, b, and c. For this reason, it is better to introduce the graphs of the equation by first taking b=c=0 and a nonzero.

The graph of the equation for any , is called a parabola with vertex the origin. It is illustrated, for several values of the constant a, in the following figure.

Figure 1

Click here to see a movie showing the change in the parabolas while a changes between 10 and -10.

The first interesting observation from Figure 1that some of the graphs are upward while some of them are downward. If we concantrate on the values of 'a' (because the only change occurs in the values of 'a'), we see that the parabola opens upward if a > 0 and downward if a <0. These are figured out in Figure 2a and Figure 2b below.

Figure 2a

Figure 2b

The second interesting observation is that to get the graph of from the graph of we stretch (in the y-direction) if and we shrink if .

The third interesting observation is that some of the pairs of graphs seem exactly the same except while one is upward the other is downward. In fact, if we consider the parabolas for and , we see that although the parabola for the former is upward, the parabola for the latter is downward. Obviously these two parabola is symmetric with respect to x-axis. If we try to look for a generalized way, we should realise that the only difference between them is that a=3 for the former and a=-3 for the latter. Therefore, and are symmetric with respect to the x-axis.

On the other hand, notice that if (x,y) satisfies , then so does (-x,y). this corresponds to the geometric fact that if the right half of the graph is reflected in the y-axis, then the left hand of the graph is obtained. We say that the graph is symetric with respect to the y-axis.

Notice that: The graph of an equation is symetric with respect to the y-axis if the equation is unchanged when x is replaced by -x.

If x and y are interchanged in , the resulting equation is , which is also represents a parabola.

To see the relations between and , let's figure out both and as the simplest case.

Figure 3 Click here to see a movie showing the change in the parabolas while a changes between 10 and -10.

A clever quess about the parabola is that it opens to the right if a > 0 and to the left if a < 0. (See Figure 4).

Figure 4a

Figure 4b

This time the parabola is symmetric with respect to the x=axis because if (x,y) satisfies , then so does (x,-y).

Notice that: The graph of an equation is symmetric with respect to the x-axis if the equation is unchanged when y is replaced by -y.

There is always certain doubts in classroom demonstrations of graphs. On of the main discussions is on the question: "Is it better to graph several graphs at once or produce them in sequence adding one at a time?"

Before giving immediate answers for this question, we should be aware of the fact that while teaching mathematics we not only want students be aware of the some mathematical facts and relations but also, more importantly, students discover these facts and relations. This idea is supported by the principles and standarts of National Council of Teachers of Mathematics (NCTM).

When we produce all of the graphs at once, students just see the whole but not parts. We can not make students to discover the facts and relations. Because, we already give the whole. Moreover, it might be difficult for most of the students to see the relations between graphs. However, when we produce graphs in sequence, we can ask questions about one graph at a time, and can ask questions about the next one and want students to produce ideas about the next graph. That leads to the discovery and problem solving learning with mathematical modelling in the students. There is lots of things to say on this topic but in sum, the answer to the above question should be " It is better to graph several graphs produce them in sequence adding one at a time instead of produce them at once".

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This page created September 18, 1999

This page last modified September 18, 1999