Assignment
3

Investigation
1

Consider
the equation ax^{2}+bx+c and to overlay several graphs of y=ax^{2}+bx+c
for different values of a, b, or c as the other two are held constant.

Answer:

We
shall hold 2 of the variables constant and allow a, b, or c to equal -3, -2,
-1, 0, 1, 2, or 3.

First,
letÕs explore a and leave b and c constant:

Equations:

á
y
= -3x^{2 }+ bx + c

á
y
= -2x^{2 }+ bx + c

á
y
= -1x^{2 }+ bx + c

á
y
= 0x^{2 }+ bx + c

á
y
= 1x^{2 }+ bx + c

á
y
= 2x^{2 }+ bx + c

á
y
= 3x^{2 }+ bx + c

Graph:

Clearly,
the negative variables for a open downward while the positive variables for a
open upward, all of the graphs pass through the point (0,1), and they all have
different vertices

Next,
letÕs explore b and leave a and c constant:

Equations:

á
y
= x^{2 }- 3x + c

á
y
= x^{2 }- 2x + c

á
y
= x^{2 }- 1x + c

á
y
= x^{2} - 0x + c

á
y
= x^{2 }+ x + c

á
y
= x^{2 }+ 2x + c

á
y
= x^{2 }+ 3x + c

Graph:

As
before, all of the graphs pass through the point (0,1) on the y-axis, open
upward, and have different vertices.

Finally,
letÕs explore c and leave a and b constant:

Equations:

á
y
= x^{2 }+ bx - 3

á
y
= x^{2 }+ bx - 2

á
y
= x^{2 }+ bx - 1

á
y
= x^{2} + bx + 0

á
y
= x^{2 }+ bx + 1

á
y
= x^{2 }+ bx + 2

á
y
= x^{2 }+ bx + 3

Graph:

Once
again all of the graphs open upwards and have different vertices. None of the
graphs share the same coordinate points, and as c increases the vertices of the
graphs shift up as well.