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Exploration #3

Graphs in the xb Plane

By Annie Sun

For equations in the following form:

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:

Notice that for the following equations and their corresponding graphs:

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:


Gives us the graph below:
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:
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:

For varying values of b and c = -1
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:

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.