# Assignment Three

# Quadratic Equations

#### Charles Meyer

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### For this investigation, I will compare different graphs of
y = x2 + bx + 1. I am concerned about how the graph changes
as values of *b* change. The values of *b* that
I really want to concentrate on are *b* = -3, -2, -1, 0,
1, 2, 3

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### The question is then posed as to what does the changes in
*b* do to the graph. It is noted that no matter what,
the graphs pass through *y* = 1 at all values of *b.*

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### The graph also helps us to see the roots of each equation.
The graph's intersection with the* x-*axis, tells the number
of roots and the value of those roots of the equation. If
*b < -*2 or *b > *2 then the equation has two
roots. Those equations when *b * is less of -2
then both roots are negative, equations where *b *is greater
than 2, then both roots are positive. Equations where *b*
is equal to -2 or 2, then the equation has only one root.
Finally *b* is greater than -2 but less than 2 then there
are no real roots to the equation.

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### Another interesting part of this graph is the vertices of
each equation. If these vertices are connected, a new parabola,
facing downward is formed.

### In this investigation, it was discovered that given an equation
of *y = ax*2* + bx + c, *if *a *and *c *remain
constant but *b* varies then the locus of the vertices of
all parabolas will also form a parabola. This investigation
would be very beneficial to a 8th or 9th grade Algebra student
in learning the value of graphing equations.

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