# 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.

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.

Continue to Page 1 (quadratic Functions - General form)

Continue to Page 2 (Find quadratic Function given its graph)

More on **quadratic functions** and related topics

Derivatives of Quadratic Functions : Explore the quadratic function *f(x) = ax *^{ 2} + b x + c and its derivative graphically and analytically.

Find Vertex and Intercepts of **Quadratic Functions** - Calculator: An applet to solve calculate the vertex and x and y intercepts of the graph of a quadratic function.

Tutorial on **Quadratic Functions** (1).

**Quadratic Functions** - Problems (1).

graphing **quadratic functions** .

**quadratic functions** in vertex form .