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The Study of Planes

By: Amanda Sawyer




In this investigation, we will study the quadratic equation in the standard Cartesian coordinate system in the xy, xa, xb, and xc plane.We will determine what effects the a, b and c variable have on each of these planes.






Letís first discuss the effects of a, b, and c in the xy plane.We can observe this change by changing each coefficient.First, letís look at what happens when the a value changes.†† The next four equations were graphed with their b and c values equaling zero.



A Value Changed in XY Plane








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Description: aXYgraph.tiff


It is clear from this picture that if our a value changes the concavity of our graph changes.When our a value is positive, we will have a graph with both endpoints going toward positive infinity.When our a values are negative, the graphs endpoints will go towards negative infinity.We also can see that the larger the absolute value of our a the more the graph is stretched.


Now letís study what happens when our b values change in our equation.The follow five equations were used to study the relationship between each equation with the a value equaling one and the c value equaling zero.†††††††††††

B Value Changed in XY Plane





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Description: bXYgraph.tiff


It is easy to see that this change in value moves our vertex for each graph, but it does not change its concavity.

Our last step in our xy plane is to study the effects of the c value.When we allow a values to equal 1 and the b values to equal 0, we will receive the following equations and graphs.

C Value Change in the XY Plane





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Description: cXYgraph.tiff


Again we can see that this changes our vertices position in our graphs.When our b = 0, the c value represents the y-intercepts of each of the five equations.From this information, we can see what characteristic of our graph have with a given a, b, or c value in our quadratic equation.


2.      THE XB PLANE

Now that we know how our graph responds in our traditional XY plane, letís study what happens when we graph equations in the XB plane.We can view this three dimensional realm plane in the picture bellow.

Description: XB.tiff


The easiest way to visualize the xb plane is to substitute b with y and graph it. We set the quadratic equal to zero to show all the possible roots of b for our equation.When we graph this equation in the two dimensional realm as in Figure 1, we can find the roots of x.When we draw a vertical line (b = 4 in green), we can find the roots of our equation from the intersection of these two lines.In the graph below we see that when we set a and c equal to one, we have two possible roots for x at b = 4. These values are located x = -.267949 and x = -3.73205.



If we graph the equation the ax + 4x + c = 0 for a and c equaling 1 in the xy plane, we get the following graph.The x-intercepts of this graph are the roots that we found in the above equation.As you can see from Figure 2, the x-intercept is the points x = -.267949 and x = -3.73205.

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We can even overlap these graphs by putting the xb-plane on top of the xy-plane:


Notice in Figure 3 how the points of intersection (green and purple lines) are exactly above the intersection of the blue graph with the x-axis.From this information we can make some generalization.Since our equation does not exist in the xb plane between -2 and 2, we do not have any x-intercepts at between those two points.




We can now repeat this for both the xa plane and the xc plane.As we can see from the table below, we can again view the roots of our equations with any given a value in the xa-plane and any given c value in the xc plane.






3-Dimensional View

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Description: CX.tiff

2-Dimensional View

Description: AXgraph.tiff

Description: CXgraph.tiff

For Solution of -3

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Description: CXgraph3.tiff

XY Plane of Equation with -3

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Description: CXgraphXY3.tiff




We can see from this information, that we can find any roots for any given a or c value.For example, we found the roots for a = -3 and c = -3 in the graphs above.Those intersections are the roots in the xy plane.††


4.     Conclusion


For all of the above problems the values of the other two variables in the plane were kept constant with a value of 1.If we change the coefficients in all the above-mentioned planes, the graphs are very difficult to view since each coefficients change in unison.This is an oversimplification because it is rare for a = b = c with parabolas, to show all the planes and their relationships to each other would require more than a mere 2 dimensions.

We can view this change in the below graph were the value for a, b, and c is equal to 0.8 in this case.In this picture xb plane is light blue, xa plane is dark gray, and xc plane is red.We can see that when your value of these variables is close to one, there isnít a root in the xa, xb, and xc plane.




In this investigation, we say the different relationships created by different planes.By using this graphing technology to create the new planes, it was easier to understand and explain these concepts to others.


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