**Department
ofMathematicsEducation**

**J. Wilson, EMAT 6680**

** **

Matthew Tanner

EMAT 6680

Write-up #2

June 27, 2003

A quadratic function is given the general form.

* *

*f(x)=ax ^{2}
+ bx + c.*

* *

Varying each
of the constants *a*,* b*, and *c*, while holding the remaining
two constant illustrates their effect on the value of the function.

Varying the
coefficient *a* has the effect of varying the breadth of the graph of the function. Let *p _{1 }*and

Since

and

.

The distance
between *p _{1}*

.

This can further be demonstrated by observing that since the second derivative the quadratic function is everywhere

the breath is

.

It is clear that the breadth of the graph of a quadratic
as defined above is characterized by *a*
alone. The effect of the
coefficient *b* is small when *x* is large.

However, proximal to the Y-axis, *b* plays a larger
role. The coefficient *b* describes the slope of the
quadratic function at the Y-axis.

Because the coefficient *b* does not affect the
breadth of the graph but does dictate the slope at the Y-axis intersection, taken
together, the effect can be seen as constraining the graph to shift along a
parabolic path.

Clickhe graph above for the demo

The *c-*coordinate has the simple effect of
shifting the graph of the function along the Y-axis.

It is
interesting to note the effect laterally shifting the graph of the function
hason the *c* -coordinate. Consider the graph of the function

.

The function has real roots
at3 and -2 as can be seen in the factorization *f(x) = (x+3)(x+2)*. The slope of the function at *x=0*
is 5 as the *b-*coefficient indicates and the Y-intercept is 6
. Now additionally consider the
function

The new function has the same breath
(second derivative) as the original function and real roots at 4 and 3 so the
effect has been to shift the graph of the original function to the left. Also note that the new function has a slope
of 7 at the Y-axis and Y-intercept of 12.
The difference in *c*-coefficients corresponds to the difference of
the Y-intercept of the original function and the value of the original function
evaluated for the value of the first derivative corresponding to the slope at
the Y-axis of the new function. It is evident
that the lateral transformation can be thought of as a composition of the
transformation brought about by a change in the *b*-coefficient and the
vertical transformation brought about by a change in the *c*-coefficient.