Question

Writing to Learn Use the concept of the derivative to define what it might mean for two parabolas to be parallel. Construct equations for two such parallel parabolas and graph them. Are the parabolas "everywhere equidistant," and if so, in what sense?

Short answer

Parabolas \(f(x) = 2x^2 + 3x + 1\) and \(g(x) = 2x^2 + 3x - 2\) are parallel because they have the same derivative, but are not 'everywhere equidistant' as the vertical distance between them varies with x.

Step-by-step solution

Understand the Concept of Derivative

A derivative of a function gives the rate of change of the function at a particular point. In the case of a parabolic function \(f(x) = ax^2 + bx + c\), the derivative \(f'(x) = 2ax + b\) gives the slope of the tangent line at any point on the parabola.

Define Parallel Parabolas

Two parabolas are considered parallel if their respective derivatives are equal, meaning they have the same slope at corresponding x-values. In other words, if we have two parabolas \(f(x) = a1x^2 + b1x + c1\) and \(g(x) = a2x^2 + b2x + c2\), they are parallel if \(f'(x) = g'(x)\) for all x, which simplifies to \(2a1x + b1 = 2a2x + b2\). This will hold true if \(a1 = a2\) and \(b1 = b2\).

Construct Equations of Parallel Parabolas

Consider an example where we have two parabolas, \(f(x) = 2x^2 + 3x + 1\) and \(g(x) = 2x^2 + 3x - 2\). These two parabolas have the same first and second coefficients, so they are parallel.

Graph the Parabolas

Graph both the equations using graphing software or manually, observing that the two parabolas do not intersect, supporting the fact that they are parallel.

Check for 'Everywhere Equidistant'

The definition of 'everywhere equidistant' would be that the distance between the two parabolas remains constant at every x-value. However, in this case our two parabolas are not 'everywhere equidistant', since the vertical distance between the two parabolas (|f(x) - g(x)|) depends on the x coordinate.