PARABOLA’S PARAMETERS EFFECTS

By Dario Gonzalez

Everyone who is familiar with functions knows that the general equation for any quadratic functions is:

The idea now is to explore the effects of each parameter  and  in the graph a quadratic function.   First, we are going to review some basic transformation when the quadratic function’s equation is written as follow:

Expression (1)

It is precise to emphasize that every quadratic function can be written as the Expression (1) by following the completion of a square method.  That is, we can go from the general expression to Expression (1) by doing as follow:

Here, we can consider:

and

Then, we obtain the Expression (1) again.  This is a very crucial fact since the Expression (1) can represent all the possible transformation of the graph a quadratic function.

Given that most of you are probably familiar with this fact, we will do a brief summary of the effects produced by parameters  and  in the graph of the quadratic function.

EFFECT OF PARAMETER :

If we consider the quadratic function where the parameters  and , and we make parameter  varies; by considering the function  as a reference, we will observe the effects of parameter a in the graph of  in the animation 1 below:

 Animation 1

From figure 1 we can conclude the effects of parameter  if we consider as a reference the function  in the following table below:

 Sign of Positive Parabola is opened upward Negative Parabola is opened downward Value of or Parabola becomes broader or Parabola becomes steep

EFFECTS OF PARAMETER :

Now we are going to consider the quadratic function equation where  and .  That is, taking as a reference the function , we can see the graphic effects of parameter h in  by observing the following animation 2:

 Animation 2

Then, we can summary the effects of parameter  in the table below:

 Sign of Positive Right move Negative Left move

EFFECTS OF PARAMETER :

Finally, if we consider a quadratic function whose parameters  and  and the function  as a reference, we visualize the graphic effect of parameter k for the function  in animation 3 below:

 Animation 3

Thus, we can summary the effects of parameter  in the table below:

 Sign of Positive Move up Negative Move down

In general, we can conclude that for each function f(x) it is possible to make the following transformations:

 Parameter Relationship Characteristic Graphic effect --------- Reflection around of X axe of f(x) graph The images of f(x) increase in a  factor The images of f(x) decrease in a  factor Positive The graph of f(x) moves right  units Negative The graph of f(x) moves left  units Positive The graph of f(x) moves up  units Negative The graph of f(x) moves down  units

Even though this explanation is enough to solve problems from 1 through 6, I would like to offer a different approach.  This approach considers all what we review before and the graphic sum.

GRAPHIC SUM:

Graphic sum consists of obtaining the graph of the sum function given the graphs of two functions.  In other words, we consider the image of each function in a fixed x, and we sum these two images obtaining one image if the sum function for that fixed x.  This concept is shown in figure 1 below:

 Figure 1

We must consider some important observations:

Observation 1: Let h(x) = f(x) + g(x), then h(x) = f(x) if g(x) = 0.

Observation 2: Let h(x) = f(x) + g(x), then h(x) is going to behave like f(x) if |g(x)| is really small.

ANOTHER APPROACH:

Consider the functions  and  whose graphs are shown in the figure 2 below:

 Figure 2

We should highlight some important issues in the sum of these two functions.

Issue 1: The sum function is going to intersect the linear function  at x = 0, according to observation 1.  Also, the sum function will behave similar to linear function close to x = 0 because, according to observation2, the images of the parabola are really small.  It is important to note that the intersection between the linear function and the sum function is tangential in x = 0.  Indeed, if we analyze the values of the functions f(x) and g(x) around x = 0, we can see the values of the sum function will be slightly larger than the linear function’s values.  The only exception occurs at x = 0 where both functions, the linear function and the sum function, are equal.

Issue 2: The images of  are going to be much larger than the images of  when the value of x goes to infinity; that is, when  the sum function will behave like a parabola because the values of the linear function will be small in comparison to the values of the parabola (observation 2).

Issue 3: Given the algebraic sum between  and  does not modify the coefficient of , we can conclude that the sum function, which is going to be another parabola, has the same breadth of , according to analysis of the parameters’ effects done above.

So, given the issues above mentioned, the graph of the sum function is going to be a parabola like , but this sum function will be tangent to the linear function  in x = 0.  The graph of the sum function is shown below:

 Figure 3

I plotted the graph of the sum function with the graphs of the others function because I would like you to appreciate how the sum graph has exactly the same shape that the parabola f(x), but it is positioned on the linear function .

Given that our above reasoning and according to observation 2 and issue 1, we can conclude that even though the green parabola could have been any parabola of the family , the sum function would have had the same behavior.  In other words, it does not matter what is the value of parameter , if we do the sum between  and , we would have obtained the same result; that is, the sum function will be a parabola like , but it would have been tangentially positioned on the linear function .

However, we can go further with this conclusion.  Observe the graphs shown in figure 4.  The blue graphs are the addend functions, and the red graph is the sum function.

 Figure 4 (a) Figure 4 (b) Figure 4 (c)

After observing these graphs, we can realize that the behavior of the sum function between a parabola  and a linear function  could be generalized in the following statement:

Statement 1: Given a parabola  and a linear function , the graph of the function h(x) = f(x) + g(x) is a parabola like , but its graph will be tangentially positioned on the linear function  in x = 0.  In other words,  is a parabola tangent to  at x = 0.

That statement establishes a crucial base for our analysis about of the effects of the parameters  and  in the graph of the quadratic function .  That is, we could consider the quadratic function h(x) as a sum of two functions  and , and from this assumption, we can draw a interesting analysis of the parameters  and .

EFFECTS OF PARAMETERS  AND :

The analysis developed above will allow us to examine the graphic effects of the parameters  and  in the quadratic function .

1) Parameter : First of all, by appreciating figure 4 (a), (b) and (c), we reason that the parameter  has two graphic representations: on the one hand,  represents the y-intersection for .  On the other hand,  determines the tangential intercept point between the linear function  and the quadratic function .

It is effortless to reach the anterior conclusion if we consider the function  as sum of  and .  We know the parameter  in the linear function g(x) represents the y-intercept, which occurs at x = 0 because .  Also, we know that , then .  Thus,  also represents the y-intersection for h(x).

Moreover, we show easily that the interception between h(x) and g(x) is tangential and occurs at .  To find the interceptions between h(x) and g(x) we must solve the equation:

So, there exist exactly one intersection between h(x) and g(x) that occurs at x = 0, and, given that , this intersection appears at .

 Figure 5

You can observe the following animation 4 to visualize graphically the effects of parameter c.

 Animation 4

2) Parameter : Recall statement 1 and issue 3, we are capable to examine the graphic effect of parameter .  The graph of a quadratic function  is just a translation of the graph of the parabola .  How we already know, h(x) is tangent to g(x) at x = 0, then  must have slope  at x = 0.  Thus, h(x) is the translation of  in a way that the point with slope  on f(x) must intercept g(x) at .

 Figure 6

If we make parameter  varies and fix parameters  and , the slope of g(x) is going to change, and the graph of g(x) is going to act as a seesaw.  So, given that h(x) is tangent to g(x) at x = 0, we can visualize the graph of h(x) seems to glide along the linear function g(x) as follow in animation 5 below:

 Animation 5

This effect of the parameter  rest upon the fact that the slope of  only depend of this parameter when .  In other words, the instantaneous change rate of h(x) is  when .  Hence,

Thus,

Therefore, the parabola h(x) moves along the linear function g(x) because the parabola h(x) has to keep tangentially touching g(x) at .

3) Parameter : We already know that the graph of a parabola  represents a translation of the graph of a parabola  such that the point on the graph of f(x) that has slope  is translated to be a tangential intersection with the liner function  at x = 0.  Thus, we can analyze the effects of the parameter  by observing the effects in , which we already did, and considering the relationship above explained between h(x) and g(x).

The parameter  modifies the breadth and the orientation of  like it was shown in figure 1.  Consider  to be positive and vary whereas we fix the other parameters  and .  As we know, the breadth of the parabola f(x) will change, and given that the statement 1 still holds, the effects of parameter  on the graph of  can be described as an up-down movement around the liner function g(x). That is, when  h(x) becomes broader and moves down around the linear function g(x).  If  h(x) becomes steep and moves up around he linear function g(x).  This is shown in the animation 6 below:

 Animation 6

Here, in figure 7 below, there is a sort of a summary picture that shows a comparison among parabolas that have different values for parameter a.  The graph reference is the graph of , which is presented by the blue parabola.

 Figure 7

Now if we consider  negative, and we have  then the parabola h(x) becomes broader and move up around the liner function g(x).  If  then the parabola h(x) becomes steep and moves down around the liner function g(x).  This is shown is animation 7 below:

 Animation 7

Here, in figure 8 below, there is a sort of a summary picture that shows a comparison among parabolas that have different values for parameter a (now for negative values).  The graph reference is the graph of , which is presented by the blue parabola.

 Figure 8

Of course,  if parameter a = 0 the parabola becomes into a linear function and, for this case, the “linear” parabola coincides with the linear function g(x) = bx + c.

I would like to highlight some final comments according to what we have been analyzing here.

Comment 1: The functions  and  have the same value for their first derivative at x = 0.

Comment 2: The function  will have a root at x = 0 if and only if c = 0.

Comment 3: The vertex of the function  will be the intersection point with the linear function  if and only if b = 0.