The Product of two Linear Functions Gives a Quadratic Function

Using basic algebra, as shown below, we can prove that the product of two linear functions gives a quadratic function. This property is explored interactively using an applet.
Two links related to the study of the properties of quadratic functions are shown below.

Let h and g be two linear functions of the form

h(x) = a x + b

and

g(x) = A x + B

where a and A are non zero constants. It can easily be shown that the product of functions h and g is a quadratic function. Let f be the function obtained as the product of g and h as follows:
f(x) = (h · g) (x) = h(x) · g(x) = ( a x + b ) · ( A x + B ) = a A x^{ 2} + (a B + b A) x + b B
An applet below may be used to explore the properties of the quadratic function f obtained above by changing the parameters a, b , A and B included in the definition of the two linear functions. There are other tutorials you may want to work through later: tutorials on quadratic functions and graphing quadratic functions.

A - Quadratic Functions From Linear Functions : Interactive Tutorial

The button below starts the applet on a separate large screen.

Click on the button above "click here to start" to start the applet and maximize the window obtained.

By default coefficients a, b, A and B are set as follows: a = 1, b = 2, A = 1 and B = 0. Explain, graphically, why the product of the two linear functions gives a function that increases indefinitely on the left side and right side.

Change coefficient A to -1. Explain, graphically, why the product of the two linear functions gives a function that decreases indefinitely on the left side and right sides.

Change all four coefficients and note that the x intercepts of the parabola are the x intercepts of the two lines. Explain.

Change all four coefficients and note that the x coordinate of the vertex of the parabola is the average of the x coordinates of the x intercepts of the parabola. Explain.

Set a, b, A and B such that: A = k a and B = k b. For example a = 1, b = 2, A = 2 a = 2 and B = 2 b = 4. The parabola has only one x intercept. Explain.