Problem 85 - Precalculus with Limits

Question

Consider the function \(f(x) = 3x^2 - 2x\). (a) Use a graphing utility to graph the function. (b) Use the \(trace\) feature to approximate the coordinates of the vertex of this parabola. (c) Use the derivative of \(f(x) = 3x^2 - 2x\) to find the slope of the tangent line at the vertex. (d) Make a conjecture about the slope of the tangent line at the vertex of an arbitrary parabola.

Step-by-Step Solution

Answer

The slope of the tangent line at the vertex of the given parabola \(3x^2 - 2x\) is 0. Based on this, one could conjecture that the slope of the tangent line at the vertex of any arbitrary parabola is always 0.

Step 1: Graph the Function

To graph \(f(x) = 3x^2 - 2x\), one would utilize graphing software or a calculator with a graphing feature. The graph of the function will look like a parabola.

Step 2: Approximate Vertex using Trace Feature

Use the trace feature on the graphing utility to approximate the vertex of the parabola. This can be done by moving along the curve until one finds the lowest point on the parabola.

Step 3: Compute the Derivative and Find the Slope at Vertex

The derivative of \(f(x) = 3x^2 - 2x\) is \(f'(x)=6x - 2\). Plug the x-coordinate from the vertex obtained from step 2 into the derivative to get the slope of the tangent line at that point. For instance, if the x-coordinate of the vertex is 'a', then the slope of the tangent line at the vertex will be \(f'(a)=6a - 2\)

Step 4: Make a Conjecture about the Slope at Vertex

After analyzing the slopes at the vertices of few functions and their graphs, one could conjecture that the slope of the tangent line at the vertex of any parabola is always 0. That is because, at the vertex, the parabola changes from curving downwards to curving upwards or vice versa, implying a horizontal (zero slope) tangent line.

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is pivotal for a comprehensive analysis of quadratic functions. The vertex is the highest or lowest point on the graph of a parabola, constituting a major feature for its graphical representation.

A parabola opens either upwards or downwards, and the vertex is at the peak for downward-opening parabolas or at the trough for upward-opening ones. Mathematically, the vertex can be found by using the formula \(h = -\frac{b}{2a}\), where \(a\) and \(b\) are the coefficients from the standard quadratic equation \(ax^2 + bx + c\). The vertex's \(y\)-coordinate can then be found by plugging the \(x\)-coordinate back into the quadratic equation.

In the context of our function \(f(x) = 3x^2 - 2x\), by using the trace feature of a graphing calculator, we approximate the vertex, which gives us a visual confirmation where the graph changes direction.

Derivative to Find Slope

The derivative of a function at a particular point provides us with the slope of the tangent line to the curve at that point. For quadratic functions, the derivative takes the form of a linear equation whose slope represents the rate of change of the function at any given \(x\)-value.

For our function \(f(x) = 3x^2 - 2x\), the derivative is \(f'(x)=6x - 2\). This represents the slope of the tangent line at any point \(x\). To find the slope of the tangent line at the vertex, we need the \(x\)-coordinate of the vertex. Once we have that, we substitute it back into the derivative formula, giving us the exact slope of the tangent at the vertex. In an ideal scenario where calculations are precise, the slope of the tangent line at the vertex for a parabola will always be 0, denoting a horizontal line.

Tangent Line at Vertex

A tangent line is a straight line that touches a curve at a single point without crossing it. At the vertex of a parabola, the tangent line represents the instantaneous rate of change of the parabola and, as a unique case, this line will always be horizontal.

This horizontal nature of the tangent line at the vertex signifies a zero slope. Why? Because at the vertex, the parabola changes its concavity – it either goes from opening upwards to increasing or from opening downwards to decreasing. Think of it as a momentarily pause in the climb or descent, which geometrically translates to a flat, horizontal line. This moment is when the slope, the rise over run, is zero given there is no 'rise' at this exact point.

For the conjecture part, it is generally true for simple quadratic functions—those without any transformation aside from scaling—that the vertex lies exactly at the point where the slope is 0. This key insight allows us to better analyze the behavior of quadratic functions at their extremities.