Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-42: Sketch the graph of the equation. $$ y=\tan ^{-1} 2 x $$

Step-by-Step Solution

Answer

The graph of \( y=\tan^{-1}(2x) \) has asymptotes at \( y = \frac{\pi}{2} \) and \( y = -\frac{\pi}{2} \), with a point at (0,0).

Step 1: Understand the Function

The given function is \( y = \tan^{-1}(2x) \), which represents the inverse tangent function applied to \( 2x \). The inverse tangent function, \( \tan^{-1}(x) \), returns the angle whose tangent is \( x \), commonly ranging from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 2: Analyze the Transformation

The function argument \( 2x \) implies that the standard arctangent function is horizontally compressed by a factor of 2. Thus, changes in \( y \) are sharper because for every unit change in \( x \), \( 2x \) changes more significantly than just \( x \).

Step 3: Identify Key Points

Determine key points by substituting values for \( x \):- When \( x = 0 \), \( y = \tan^{-1}(0) = 0 \).- As \( x \to \infty \), \( 2x \to \infty \) and \( y \to \frac{\pi}{2} \).- As \( x \to -\infty \), \( 2x \to -\infty \) and \( y \to -\frac{\pi}{2} \).

Step 4: Sketch the Graph

Start by plotting the key points: (0, 0) on the graph. Draw asymptotic lines approaching \( y = \frac{\pi}{2} \) as \( x \to \infty \) and \( y = -\frac{\pi}{2} \) as \( x \to -\infty \). Connect these points smoothly to depict the horizontal compression.

Key Concepts

Inverse Tangent Function

The inverse tangent function, often written as \( \tan^{-1}(x) \) or \( \arctan(x) \), is a fundamental concept in trigonometry. It essentially helps us find the angle whose tangent value is \( x \). This function is particularly useful because it allows us to transition between the tangent value and the angle itself. In its traditional form, the output of \( \tan^{-1}(x) \) is an angle between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians.

• It captures the relationship between an angle in a right triangle and the ratio of the opposite to the adjacent sides.
• This function is critical when the need arises to unpack the tangent values in computations and data interpretations.

It's a key element not just in trigonometry but also in calculus and other areas that require insight into angular relationships from values.

Arctangent Transformation

Transformations occur often in function graphing to change the appearance and properties of the original function. With the function \( y = \tan^{-1}(2x) \), an arctangent transformation has taken place. But what exactly changes?

• The transformation primarily applies a horizontal compression to the function.
• Instead of evaluating \( \tan^{-1}(x) \), we're now evaluating \( \tan^{-1}(2x) \).
• This implies that the input \( x \) gets multiplied by 2, meaning it reaches the same tangent value but quicker along the x-axis.

Such transformations reflect how modifications within function arguments affect their graphical representations. Recognizing these changes can greatly aid in quick and accurate graph sketching.

Graph Sketching

Sketching the graph of a transformed arctangent function like \( y = \tan^{-1}(2x) \) can be straightforward once you understand key transformations and behaviour.

• Start by identifying key points, such as where \( x=0 \). Here, \( y = \tan^{-1}(0) = 0 \).
• Next, consider the behavior as \( x \to \infty \) and \( x \to -\infty \). These limits help find horizontal asymptotes.
• Since the original arctangent graph rises to \( \frac{\pi}{2} \) and falls to \(-\frac{\pi}{2} \), compressions affect how fast the graph reaches these values.

Plot these points to establish the curve, and ensure you draw smooth transitions that capture these features. The result is a graph that visually displays the horizontal compression compared to a standard arctangent graph.

Horizontal Compression

Horizontal compression is a graphical concept involving the 'squeezing' of a function across the x-axis. For the function \( y = \tan^{-1}(2x) \), this compression occurs because of the \( 2x \) term.

• It means that inputs change more rapidly, effectively compressing the wave pattern of the arctangent curve.
• This transformation alters the rate at which \( y \) valuations occur as \( x \) grows positively or negatively.
• Compression by a factor of 2 indicates that any point previously at \( x \) is now at \( \frac{x}{2} \), making the graph "narrower" horizontally.

Understanding this helps accurately project how the function's graph will look in comparison to its uncompressed counterpart. Such insights are crucial for identifying function behavior quickly and effectively.