Exploration of the Quadratic Equation

By Brooke Norman

 

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In this assignment, we are going to look at the quadratic equation of:

 

 

 

I will use Graphing Calculator 3.2 to examine the effects of changing the values for a, b, and c.

 

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First, lets explore the equation by letting b and c equal 1 and varying a from -3 to 3. 

        

 

As you can see from the above graphs, each equation crosses the y-axis at 1.  The graph when a=0 is a straight line that is tangent to the other graphs.  When a is a positive number, the parabolas are concave up and when a is negative, the parabolas are concave down. As a changes, so do the vertexes. As the value of a increases, the parabolas become more steep and skinnier.   

 

 

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Now, lets explore the equation but this time lets let a and c equal to 1 and let b vary from -3 to 3.

 

 

 

 

The graph shifts to the left as b increases in a positive direction and the graph shifts to the right as b increases in a negative direction. Notice that the vertex shifts as b is varied.

 

 

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We are now going to explore the equation but this time lets let a and b equal to 1 and let c vary from -3 to 3.

     

 

As you see, when c varies, the vertex is shifting up and down.  Therefore c is changing the y-coordinate of the vertex.

 

 

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In conclusion:

The position of the vertex is affected by a, b, and c.  The x- and y-coordinates of the vertex are affected by a and b and only the y-coordinate appears to be affected by c.  It also appears that a also determines the steepness and he direction of concavity of the parabola.

 

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