Problem 19 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of horizontal stretches and compressions to graph at least two cycles of the given functions. $$f(x)=\tan (2 x)$$

Step-by-Step Solution

Answer

To graph the function \( f(x)=\tan (2 x) \), start by identifying that the period of the function is \( \pi/2 \). From there, identify the key points for one cycle at \( x=-\pi/4, 0, \pi/4 \) then sketch the curve of the tangent using these points. Use the same process to sketch additional cycles by adding the period to each x-value. Repeat these steps for as many cycles as required.

Step 1: Identify the Period

The period of the generic tangent function \( \tan(x) \) is \( \pi \). However, by multiplying 'x' by 2 in \( \tan(2x) \), the period of the function is halved. Therefore, the period of \( f(x)=\tan (2 x) \) is \( \pi/2 \).

Step 2: Identify Key Points for One Cycle

For one cycle of the tangent function we usually take key x-values as : \( -\pi/2, 0, \pi/2 \). But as there is a 2 multiplier to the x, those points become: \( -\pi/4, 0, \pi/4 \). Here, the tangent function approaches \( -\infty \) as it approaches \( -\pi/4 \), it equals 0 at \( x=0 \), and approaches \( +\infty \) as it approaches \( \pi/4 \) from the left.

Step 3: Sketch the First Cycle

Draw a graph based on the key points identified in step 2. Draw asymptotes (lines that the function approaches but never crosses) at \( x=-\pi/4 \) and \( x=\pi/4 \). Mark points at \( x=0 \), and draw the curves of tangent which start from \( -\infty \) at \( x=-\pi/4 \), pass through (0,0), and head towards \( \infty \) at \( x=\pi/4 \).

Step 4: Sketch the Second Cycle

To plot the next cycle, repeat this process but add the period \( \pi/2 \) to each x-value. Thus, key points for second cycle will be at \( \pi/4, \pi/2, 3\pi/4 \)

Step 5: Repeat the Process

These processes can be repeated as many times as necessary to sketch additional cycles.

Key Concepts

Horizontal Stretches

Understanding the concept of horizontal stretches can make it easier to transform and graph trigonometric functions. A horizontal stretch occurs when all the x-values of a function are multiplied by a factor, making the graph "stretch" horizontally.

Imagine you have a rubber band and you pull it outward in both horizontal directions; this action is similar to a horizontal stretch on a graph.

This transformation affects the original x-values, specifically stretching them apart by a certain factor. In the context of trigonometric functions, the key here is that every x-value in the graph gets multiplied or divided by the stretch factor.

• For example, if you have a function like \( f(x) = \tan(ax) \), a horizontal stretch occurs for values of \( a \) less than 1.
• This means the graph will be stretched to cover a larger interval for each cycle of the function.

In practice, identifying stretches directly impacts how and where we plot key points and asymptotes on a graph. Always make sure to note the factor of change, as this determines your new key x-values.

Horizontal Compressions

Horizontal compressions are the counterpart to horizontal stretches and represent situations where the graph of a function "compresses" together along the horizontal axis.

If you take the same rubber band analogy, this time you are squeezing it inward towards the center. This process reduces the distance between the x-values, effectively compressing the graph of the function.

In trigonometric functions, understanding compressions can be essential for accurate graph plotting.

• With a function like \( f(x) = \tan(2x) \), the graph undergoes a horizontal compression.
• This means that each complete cycle of the function, which typically takes \( \pi \) radians for \( \tan(x) \), will compress to fit into \( \pi/2 \) radians.

Simply put, the factor that compresses the function is reflected in the expression \( \tan(ax) \), where \( a > 1 \) results in a horizontal compression. Understanding this compression helps predict where each cycle ends and begins, thus aiding your graphing process.

Period of a Function

The period of a function is a critical concept for interpreting and graphing periodic functions like trigonometric ones. The period refers to the horizontal length of one complete cycle of the given function.

For a basic tangent function \( \tan(x) \), the period is \( \pi \). This means it takes \( \pi \) units along the x-axis for one full cycle of the tangent curve to complete.

However, when transformations happen, such as multiplying the x-variable, the period changes. This is seen with the function \( \tan(2x) \), where the period becomes \( \pi/2 \).

• This change occurs because multiplying the x-variable by a factor effectively divides the original period by that factor.
• Therefore, every cycle of \( \tan(2x) \) completes twice as fast as \( \tan(x) \), compressing the wave.

It's essential to identify the period of a function to correctly plot it over multiple cycles. Knowing how transformations affect the period allows you to predict the graph's oscillations along the x-axis.