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b. Given that the power lost in a resistive element is the product of
the voltage across the resistor multiplied by the current through
the resistor, plot the power through the variable resistor as a func-
tion of R.
c. Determine the value of R such that the power lost in this resistor
is maximized.
d. Find, in general, the relation between R and R that ensures that
1
the power lost in the load resistance is maximized. (This general
result is called Thevenin’s theorem.)

3.3 Examples with Quadratic Functions
A quadratic function is of the form:

y(x) = ax + bx + c (3.8)
2

Preparatory Exercises
Pb. 3.9 Find the coordinates of the vertex of the parabola described by Eq.
(3.8) as functions of the a, b, c parameters.
Pb. 3.10 If a = 1, show that the quadratic Eq. (3.8) can be factored as:

y(x) = (x – x )(x – x )
+
–
where x are the roots of the quadratic equation. Further, show that, for arbi-
±
c −b
trary a, the product of the roots is , and their sum is .
a a

In-Class Exercises
Pb. 3.11 Develop a function M-file that inputs the two real roots of a second-
degree equation and returns the value of this function for an arbitrary x. Is
this function unique?
Pb. 3.12 In your elementary mechanics course, you learned that the trajec-
tory of a projectile in a gravitational field (oriented in the –y direction) with

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