Assignment3

investigation 1

I look for the coordinates of the vertce of the parabora,

So, the coordinates of the vertice is .

I make an equation by using this simultaneous equqtions.

At first, I change (1).

Next, substitute (3) for (2)

This is the result desired, the locus of the vertices.

In general, the locus of the vertices of is

investigation2

Here are the graphs (Purple) and(red)

It is clear that, for any b, has two intersection points with a horizontal line. This means that we get two real roots of , one is positive and the another is negative.

we can calucurate this roots as below,

that are always real numbers.

I show 7 graphs for 7 values of c as -3, -2, -1, 0, 1, 2, 3.

When c is positive, the graph exists the 2nd quadrant and the 4th quadrant separately.

When c is negative, the graph exists on the left of b-axis(the area where x < 0) and the right of b-axis(the area where x > 0) separately.

When c = 0, the graph is a line throuh the origin.

investigation3

I draw the graphs,, , and on the same axex.

The graphhas two intersection points with the graph (red) and no intersection point with the graph(purple).

This means that there are two sets(x, b coordinates) of real roots for the simultaneous equations.

(and that there is no real root for the set of equations.)

The two sets of real roots are (x, b) = (1, -2) and (-1, 2). These are the points where the graphs intersect.

investigation4

On xc plane, I show 7 graphs of for 7 values b as -3, -2, -1, 0, 1, 2, 3.

All of the graphs are going through the origin.

The locus of vertices is a parabora.

Here I show you the animation of the moving graph ( b changes from -5 to 5).

when b = 0, we get one root of the equation with the line c = 0.

When b is not equal to 0, we get two roots of the equation with the line c = 0.

investigation5

On xa plane, I show 7 graphs of for 7 values b as -3, -2, -1, 0, 1, 2, 3.

Where x = 0, the graph does not exist. In this case, there is no root of the equation with the line a = 0.

For any b, when the value of x goes to the infinity or negative infinity, the value of a approaches to 0.

Here I show you the animation of the moving graph ( b changes from -5 to 5).

The graphs have two parts, the right part and the left part.

The right and left graph are same shape when b = 0.

The right part is shaped like a mountain when b is negative.

The left part is shaped in this way when b is positive.

For any b except for b = 0, we get exact one root of the equation with the line a = 0.

Investigation6

On xb plane, I construct a movie of (a changes from -5 to 5).

The number of roots of this equation with the line b = 0 is one, two, or three.

Can you see the graph having three roots?

Next, on xc graph, I show a movie of (a changes from -5 to 5.).

The number of roots of this equation with the line c = 0 is 0 (when a = 0) or one.

On xd graph, I make a movie of (a changes from -5 to 5).

All the graphs are go through the origin (0, 0).

The number of roots of this equation with the line d = 0 is one, two(when a = 0), or three.