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Exploration #3

Graphs in the xb Plane

By Annie Sun


For equations in the following form:
assign3_1

Here are graphs with varying values of b (-5, 5), with a = 1 and c = 1

assign3_2 assign3_3

All the graphs pass through the point (0,1) and the locus of the vertices was found using (0,1) as the vertex with the equation:
assign3_4
assign3_5


Notice that for the following equations and their corresponding graphs:
assign3_6assign3_7

When b = 2 and b = -2, there is one negative and one positive real root, respectively.  When -2 < b < 2, the equation has no real roots. 


Now let’s examine the equations in the xb plane:

Graphing
 assign3_8

Gives us the graph below:
assign3_9
Notice the gap between b = -2 and b = 2, this corresponds to the previous statement of when -2 < b < 2, there are no real roots.


Let’s use b = 3 in the graph below:
assign3_10
We see that the horizontal line intersects the graph at two points that correspond to the roots of the original equation in the xy plane; in particular assign3_11 and its two negative real roots.


Now let’s take a look at other values for c in the following equations and their corresponding graphs:
assign3_12assign3_13

For varying values of b and c = -1
assign3_14assign3_15
We can see from the graph that each equation has two real roots which correspond to the b values (-2,2) that intersect the graph at two points below:
assign3_16

We can use different values for c:
assign3_17 assign3_18

And the corresponding roots to the original equations are found where the y=b line cross the graph.


 


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