Assignment 3

Quadratic Equations

 

Page Bird

 

Investigation 1.


Consider the locus of the vertices of the set of parabolas graphed from

Show that the locus is the parabola

 


 



In this graph, the values of b range from 3 to  3.

 


The parabolas all intercept at the point (0,1).

Using the graphing calculator, it is clear that the vertices of the parabolas graphed above are

(1.5, -1.25), ( -1, 0), ( -.5, .75),  (0, 1), ( .5, .75), ( 1, 0), 

(1.5, -1.25).

 

Using slope and y-intercept form, the equations of the lines that go through points (1, 0), ( -1, 0), and (0, 1) are

 

 
and

 


Interestingly,


The graph below shows that the product of the equations of the two lines is a parabola. The vertices of each parabola with the equation


 maps to the graph below.


 

 

 


To view a GSP file of this parabola click here.

 

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