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)=ax2 + 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 p1 and p2 be the values of  x  where the slope of the function is 1 and 1 respectively.  That is, f(p1)=-1 and  f(p2)=1.  These are two arbitrarily chosen but easily recognizable landmarks on any graph of a quadratic function.

Since

and

.

The distance between p1 and p2 is

.

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 at3 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.