Quadratic Functions from the Product of Two Linear Functions

Using fundamental algebra, we can prove that multiplying two linear functions produces a quadratic function. This relationship is explored interactively below.

Let \( h \) and \( g \) be two linear functions defined as:

\[ h(x) = a x + b \] \[ g(x) = A x + B \]

where \( a \) and \( A \) are non-zero constants. The product of these functions yields a quadratic function:

\[ f(x) = (h \cdot g)(x) = h(x) \cdot g(x) = (a x + b)(A x + B) \]

Expanding the product:

\[ f(x) = aA x^{2} + (aB + bA)x + bB \]

Adjust the coefficients below to see how the graphs change in real-time. Note how the x-intercepts of h(x) and g(x) become the x-intercepts of f(x).

Interactive Graph

h(x) = 1.00x + 2.00
g(x) = 1.00x + 0.00
f(x) = h(x) × g(x) = 1.00x² + 2.00x + 0.00
X-Intercepts:
h(x) = 0 when x = -2.00
g(x) = 0 when x = 0.00
f(x) = 0 when x = -2.00, 0.00
Hover over graph

Observations & Analysis

  1. X-Intercept Inheritance: The quadratic function f(x) is zero exactly where either h(x) or g(x) is zero. This is because if h(x) = 0 or g(x) = 0, then their product f(x) = h(x)·g(x) = 0.
  2. Y-Intercept: f(0) = h(0)·g(0) = b·B. The y-intercept of the parabola is the product of the y-intercepts of the lines.
  3. Vertex Position: The vertex of the parabola lies exactly halfway between the two x-intercepts when they are distinct.
  4. Double Root: When h(x) and g(x) share the same x-intercept (i.e., when \(-\frac{b}{a} = -\frac{B}{A}\)), the parabola has a single x-intercept (a double root).
  5. Parabola Opening: The parabola opens upward if a·A > 0 and downward if a·A < 0.

Further Study on Quadratic Functions