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Vary the coefficient b
in a
vertex trace of Y=ax2+bx+c
SOLUTION
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View a Geometer's SketchPad file with
this construction
Conjecture
The red path above suggests that the vertex trace will
form a quadratic function (parabola) when you vary the coefficient
b in the equation Y=ax2+bx+c. We can verify
this conjecture by finding an association beteen the x and y-coordinates
of a vertex of a parabola.
Method
You can find the coordinates of the vertex using techniques
in advanced algebra, such as putting the equating into vertex
form after completing the square. However, this method is effective
for only this investigation. If you examine the trace of extrema
of a cubic function, for example, you will need to rely on differential
calculus. Therefore, in an effort to remain consistent in all
of the solutions, I will use differential calculus as a means
of finding relative extrema, such as the vertex of a parabola.
The vertex of a parabola is a relative maximum or minimum. In differential calculus, you can detect the existence of a relative maximum or minimum by identifying the location where the derivative changes sign.
Location of Vertex
In this case, we have:
Take the derivative and set it equal to zero.
Therefore, the vertex of the parabola (maximum or minimum) will
occur when
The y-coordinate of the vertex can be found by substituting this
x-value back into the function.
simplifies to
Therefore, the vertex of any parabola will be located at
Proof
Any trace of the vertex must relate the x and y-coordinates
in a true statement. If the coefficient b varies, then
the locus will be the quadratic function
because
This is part of the solution to the vertex trace of a quadratic
function investigatation. You can explore this phenomenon by playing
with the interactive applet and
answering the questions that follow.
Last revised: July 7, 2000