Key Features Of H(x) = 2 Tan(x) - 1 Range And Period Analysis

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Introduction

In the fascinating world of trigonometry, the tangent function holds a special place. It's a periodic function with unique characteristics that set it apart from its sinusoidal cousins, sine and cosine. When we introduce transformations to the basic tangent function, such as stretching, shifting, and reflecting, the resulting graphs exhibit a rich variety of behaviors. In this article, we will delve into the key features of the function h(x) = 2 tan(x) - 1. This function is a transformed version of the basic tangent function, and by carefully examining its equation, we can uncover its range, period, midline, and other essential properties. Understanding these features is crucial for accurately graphing the function and for applying it in various mathematical and scientific contexts. We will explore statements about the range and period of this function, determining which are true and providing a comprehensive explanation.

Understanding the Tangent Function

Before diving into the specifics of h(x) = 2 tan(x) - 1, let's take a moment to review the fundamental characteristics of the tangent function, tan(x). The tangent function is defined as the ratio of the sine function to the cosine function: tan(x) = sin(x) / cos(x). This definition gives rise to several key properties:

  • Periodicity: The tangent function is periodic, meaning its graph repeats itself over regular intervals. The period of the basic tangent function is Ï€, which is significantly different from the period of sine and cosine, which is 2Ï€. This periodicity arises from the periodic nature of both sine and cosine, but the division introduces a repeating pattern over a shorter interval.
  • Vertical Asymptotes: The tangent function has vertical asymptotes at values of x where the cosine function is zero. This occurs at x = (2n + 1)Ï€/2, where n is any integer. At these points, the tangent function is undefined, and the graph approaches infinity (or negative infinity). These asymptotes are a defining feature of the tangent function's graph.
  • Range: The range of the tangent function is all real numbers, meaning it can take on any value from negative infinity to positive infinity. This is because the sine and cosine functions oscillate between -1 and 1, and their ratio can take on any value depending on their relative magnitudes.
  • Symmetry: The tangent function is an odd function, which means that tan(-x) = -tan(x). This symmetry is reflected in the graph of the tangent function, which is symmetric about the origin.

These properties of the basic tangent function form the foundation for understanding the behavior of transformed tangent functions like h(x) = 2 tan(x) - 1. By recognizing how transformations affect these fundamental characteristics, we can accurately predict and interpret the graphs of more complex tangent functions.

Analyzing h(x) = 2 tan(x) - 1

Now, let's focus on the function h(x) = 2 tan(x) - 1 and dissect its key features. This function is a transformation of the basic tangent function, and the coefficients in the equation reveal how the graph has been modified. The '2' in front of the tangent function represents a vertical stretch, while the '-1' represents a vertical shift. Understanding these transformations is essential for determining the range, period, and midline of h(x).

A. The Range of the Function

The range of a function refers to the set of all possible output values (y-values) that the function can produce. For the basic tangent function, tan(x), the range is all real numbers. This is because the tangent function can take on any value from negative infinity to positive infinity as x varies. The vertical stretch in h(x) = 2 tan(x) - 1 does not affect the range, as it simply multiplies the output values by 2, which still spans all real numbers. The vertical shift of '-1' also doesn't change the range; it only shifts the entire graph down by 1 unit, but the function still extends infinitely in both the positive and negative y-directions. Therefore, the range of h(x) = 2 tan(x) - 1 remains all real numbers. This makes Statement A true. To solidify this understanding, consider that no matter how large or small a real number you choose, you can always find an x-value that will produce that output in h(x). The tangent function's ability to reach any real number, combined with the properties of vertical stretches and shifts, ensures that the transformed function maintains this characteristic.

B. The Period of the Function

The period of a trigonometric function is the length of one complete cycle before the graph repeats itself. For the basic tangent function, tan(x), the period is π. This means that the graph of tan(x) repeats every π units along the x-axis. Transformations such as vertical stretches and shifts do not affect the period of the tangent function. The vertical stretch in h(x) = 2 tan(x) - 1 changes the steepness of the graph but doesn't alter how frequently it repeats. Similarly, the vertical shift simply moves the graph up or down without affecting its periodicity. The period is only affected by horizontal stretches or compressions, which are determined by the coefficient of x inside the tangent function's argument. In this case, the coefficient of x is 1, so the period remains π. Therefore, the period of h(x) = 2 tan(x) - 1 is π, not π/2. This makes Statement B false. Understanding this concept is crucial for accurately graphing tangent functions and predicting their behavior over different intervals.

Determining the True Statements

After carefully analyzing the function h(x) = 2 tan(x) - 1, we can definitively determine which statements are true. We've established that:

  • Statement A: The range of the function is all real numbers. This statement is true because the vertical stretch and shift do not limit the possible output values of the function. The tangent function's inherent range, which spans all real numbers, is preserved under these transformations.
  • Statement B: The period of the function is Ï€/2. This statement is false. The period of the tangent function is only affected by horizontal stretches or compressions. Since there is no horizontal transformation in h(x) = 2 tan(x) - 1, the period remains the same as the basic tangent function, which is Ï€.

Therefore, only Statement A is true about the key features of the graph of the function h(x) = 2 tan(x) - 1.

Additional Insights and Applications

Understanding the transformations of trigonometric functions, like the tangent function, is not just an academic exercise. It has significant practical applications in various fields, including:

  • Physics: Tangent functions are used to model phenomena involving angles and ratios, such as the angle of elevation in projectile motion or the relationship between the sides of a right triangle in optics.
  • Engineering: Engineers use tangent functions in designing structures, calculating slopes, and analyzing oscillatory systems. Understanding the period and range of these functions is crucial for ensuring stability and predictability in engineering designs.
  • Computer Graphics: Tangent functions play a role in creating realistic perspectives and transformations in computer graphics. Understanding how to manipulate these functions allows for the creation of complex and visually appealing images.

By mastering the concepts of range, period, and transformations, you gain a powerful tool for analyzing and applying trigonometric functions in a wide range of contexts. This knowledge not only enhances your mathematical understanding but also opens doors to practical problem-solving in various scientific and technological domains. For instance, consider a scenario where you need to model the height of a shadow cast by a tall building at different times of the day. The tangent function, with appropriate transformations to account for the building's height and the angle of the sun, could provide an accurate representation of the shadow's length throughout the day. Similarly, in signal processing, understanding the periodic nature of trigonometric functions is essential for analyzing and manipulating waveforms.

Conclusion

In conclusion, by carefully analyzing the function h(x) = 2 tan(x) - 1, we've determined that the range of the function is all real numbers (Statement A) is the true statement. The period of the function is π, not π/2 (Statement B). Understanding the key features of transformed trigonometric functions like this one is crucial for a deeper understanding of trigonometry and its applications in various fields. By recognizing the effects of vertical stretches and shifts on the range and period, we can accurately interpret and predict the behavior of these functions. This knowledge empowers us to solve a wide range of mathematical and real-world problems, from modeling physical phenomena to designing engineering structures.