Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Sketch the graph of \(g(x)=\tan 2 x\) on \([0,2 \pi]\).

Step-by-Step Solution

Answer

The graph of \(g(x)= \tan 2x\) over the range [0, 2π] has 4 complete cycles of the tangent function, each of period π/2 and ranging from negative to positive infinity, with asymptotes at even multiples of \(\pi/4\).

Step 1: Understand the Standard Tangent Function

Begin by remembering what the graph of the standard \(\tan x\) looks like. It has a period of \(\pi\), asymptotes where the function is undefined at odd multiples of \(\pi/2\), and it passes through the origin.

Step 2: Adjust for the '2x'

The '2x' inside the \(\tan\) function means that every 'x' is effectively doubled, meaning that everything on the \(\tan\) graph will happen 'twice as fast'. Specifically, this halves the period of the function from \(\pi\) to \(\pi/2\), and the asymptotes will now occur at multiples of \(\pi/4\). In each new period, the function will rise from negative infinity at the first asymptote, through 0 at the mid-point, to positive infinity at the next asymptote.

Step 3: Sketch the Graph

Now, we can start to sketch the function. Over the range \([0, 2\pi]\), because of the halved period, there will now be four complete 'cycles' of the tangent function. At \(x = 0\), draw an asymptote. The function rises from negative infinity to positive infinity as 'x' varies from \(0\) to \(\pi/4\), passes through the origin at \(x=\pi/8\). Repeat this pattern for the ranges \((\pi/4, \pi/2)\), \((\pi/2, 3\pi/4)\), and \((3\pi/4, \pi)\). This covers the first half of the given range. Repeat the entire pattern for the range \((\pi, 2\pi)\) completing the sketch of the function over the entire range.

Key Concepts

Tangent Function Period

When graphing the tangent function, one of the fundamental attributes to understand is its period. The period of a function is the interval length over which the function repeats its shape. For the standard tangent function, denoted as \( y = \tan x \), the period is \( \text{π} \). This means that every \( \text{π} \) units along the x-axis, the tangent curve starts its pattern anew.

However, when we introduce a coefficient before the variable x, like in \( g(x) = \tan 2x \), it alters the period of the function. Specifically, the new period can be found by dividing the standard period \( \text{π} \) by the absolute value of this coefficient. Hence, for \( g(x) = \tan 2x \), the period becomes \( \text{π} / 2 \). When preparing to sketch the graph of a transformed tangent function, noting the new period is a crucial first step as it dictates the horizontal stretch or compression of the graph.

Tangent Asymptotes

Tangent asymptotes are vertical lines that the graph of a tangent function approaches but never crosses or touches. These lines represent values for which the tangent function is undefined. In the case of the standard tangent function \( y = \tan x \), asymptotes occur at odd multiples of \( \text{π}/2 \), such as \( \text{π}/2, 3\text{π}/2, -\text{π}/2 \), and so on.

When dealing with the function \( g(x) = \tan 2x \), the locations of these asymptotes change. Since the period is now \( \text{π}/2 \), the asymptotes will be found at each value that makes the inside of the tangent function (2x in this case) equal to an odd multiple of \( \text{π}/2 \). Therefore, for \( g(x) = \tan 2x \), asymptotes will appear at intervals of \( \text{π}/4 \) since \( 2x = (2n+1)\text{π}/2 \) leads to \( x = (2n+1)\text{π}/4 \), where \( n \) is any integer. Acknowledging the location of these asymptotes is vital for correctly sketching the behavior of the tangent function around these vertical lines.

Graph Transformations

Graph transformations are changes that affect the original position and shape of a graph. These can include translations (shifting the entire graph vertically or horizontally), reflections (flipping the graph over an axis), and scalings (stretching or compressing the graph in the vertical or horizontal direction).

Effect of Coefficients and Addition

In the context of the tangent function such as \( g(x) = \tan 2x \), the coefficient '2' in front of 'x' is a horizontal scaling. It compresses the graph horizontally by a factor of 1/2. This means that each cycle of the tangent's wave occurs in half the space on the x-axis compared to the standard tangent function. Having a solid grasp of how these transformations alter the graph will aid students in sketching and interpreting trigonometric functions.

Sketching the Transformed Tangent

When sketching \( g(x) = \tan 2x \), one must account for how the period and asymptotes have changed to ensure accuracy. Moreover, understanding the basic shape of a single cycle of the tangent function - starting at an asymptote and rising to infinity, passing through zero, and then descending to negative infinity to meet the next asymptote - is crucial. This repeating pattern, combined with knowledge of the function's period and asymptotes, allows for a precise representation of the transformed tangent function.