Range Of Y Equals -5sin(x) A Comprehensive Explanation

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In mathematics, particularly in trigonometry and calculus, understanding the range of trigonometric functions is crucial for solving a variety of problems. The range of a function refers to the set of all possible output values (y-values) that the function can produce. In this article, we will delve into determining the range of the function y = -5sin(x). This function is a transformation of the basic sine function, and understanding these transformations is key to finding the range. We will explore the properties of the sine function, how vertical stretches and reflections affect the range, and finally, arrive at the correct answer.

Before we dive into the specifics of y = -5sin(x), let's first establish a solid understanding of the basic sine function, y = sin(x). The sine function is a periodic function, which means its values repeat over regular intervals. Graphically, this is represented by a wave that oscillates between certain maximum and minimum values. The sine function’s graph starts at zero, rises to a maximum value, returns to zero, falls to a minimum value, and then returns to zero again, completing one full cycle.

Key Properties of y = sin(x):

  • Amplitude: The amplitude of a sine function is the distance from the midline (the horizontal line that runs through the center of the wave) to the maximum or minimum value. For y = sin(x), the amplitude is 1. This means the maximum value the function reaches is 1, and the minimum value is -1.
  • Period: The period is the length of one complete cycle of the wave. For y = sin(x), the period is 2Ï€, meaning the pattern repeats every 2Ï€ units along the x-axis.
  • Range: The range of y = sin(x) is the set of all possible y-values. Since the sine function oscillates between -1 and 1, the range is -1 ≤ y ≤ 1. This means that for any value of x, the output of sin(x) will always be a value between -1 and 1, inclusive.

Visualizing the graph of y = sin(x) can be incredibly helpful. Imagine a smooth wave oscillating between the lines y = -1 and y = 1. The function starts at (0, 0), rises to a peak at (Ï€/2, 1), returns to zero at (Ï€, 0), reaches a trough at (3Ï€/2, -1), and completes its cycle back at (2Ï€, 0). This cyclical behavior and the defined boundaries of -1 and 1 are crucial for understanding the sine function's range.

Understanding these basic properties allows us to analyze how transformations of the sine function affect its range. Transformations such as vertical stretches, compressions, and reflections can alter the amplitude and position of the sine wave, thereby changing the set of possible y-values. In the next section, we will explore how multiplying the sine function by a constant, as in y = -5sin(x), impacts its range.

Now that we have a firm grasp on the basic sine function, y = sin(x), let's examine how transformations affect the range. Specifically, we will focus on vertical stretches and reflections, which are crucial for understanding the range of y = -5sin(x). Transformations involve altering the function's equation, which in turn changes the graph and, consequently, the function's range.

Vertical Stretch:

A vertical stretch occurs when we multiply the sine function by a constant greater than 1. For example, consider the function y = Asin(x), where A is a constant. If A > 1, the graph of the sine function is stretched vertically. This means that the amplitude of the wave increases, and the function's maximum and minimum values move further away from the x-axis.

In the case of y = 5sin(x), the constant A is 5. This means that the graph of y = sin(x) is stretched vertically by a factor of 5. The amplitude of the transformed function is now 5, so the maximum value is 5, and the minimum value is -5. Therefore, the range of y = 5sin(x) is -5 ≤ y ≤ 5.

Reflection Across the x-axis:

A reflection across the x-axis occurs when we multiply the sine function by -1. This transformation flips the graph vertically. The parts of the graph that were above the x-axis are now below, and vice versa.

For instance, consider the function y = -sin(x). This function is a reflection of y = sin(x) across the x-axis. The maximum value of y = sin(x) (which is 1) becomes the minimum value of y = -sin(x) (which is -1), and the minimum value of y = sin(x) (which is -1) becomes the maximum value of y = -sin(x) (which is 1). However, the range remains the same: -1 ≤ y ≤ 1, although the function's behavior is inverted.

Combining Vertical Stretch and Reflection:

The function y = -5sin(x) combines both a vertical stretch and a reflection. The multiplication by 5 stretches the graph vertically, making the amplitude 5, and the multiplication by -1 reflects the graph across the x-axis. This combined transformation alters both the amplitude and the direction of the wave.

To visualize this, start with y = sin(x), which has a range of -1 ≤ y ≤ 1. Stretching it vertically by a factor of 5 gives y = 5sin(x), which has a range of -5 ≤ y ≤ 5. Finally, reflecting it across the x-axis results in y = -5sin(x). The reflection does not change the range; it only inverts the function's behavior. Therefore, the range of y = -5sin(x) is still -5 ≤ y ≤ 5.

Understanding these transformations helps us to analyze more complex trigonometric functions. By breaking down the transformations step by step, we can accurately determine how they impact the range of the function. In the next section, we will apply this understanding to identify the correct range for y = -5sin(x) from the given options.

Now, let's apply our understanding of sine function transformations to determine the range of the function y = -5sin(x). We've already established that the sine function, y = sin(x), has a range of -1 ≤ y ≤ 1. The function y = -5sin(x) is a transformation of the basic sine function, involving both a vertical stretch and a reflection across the x-axis.

Step-by-Step Analysis:

  1. Vertical Stretch: The multiplication by 5 in y = -5sin(x) means the sine function is stretched vertically by a factor of 5. This changes the amplitude from 1 to 5. So, the possible y-values now range from -5 to 5.
  2. Reflection Across the x-axis: The negative sign in y = -5sin(x) indicates a reflection across the x-axis. This reflection inverts the graph, but it does not change the range. The maximum and minimum values are still 5 and -5, respectively, but the function's direction is flipped.

Combining the Transformations:

The vertical stretch by a factor of 5 expands the range to -5 ≤ y ≤ 5. The reflection across the x-axis does not alter the range; it only changes the orientation of the sine wave. Therefore, the range of y = -5sin(x) remains -5 ≤ y ≤ 5.

Analyzing the Answer Choices:

We are given the following answer choices:

  • A. All real numbers
  • B. -5 ≤ y ≤ 5
  • C. All real numbers -5/2 ≤ y ≤ 5/2
  • D. All real numbers -1 ≤ y ≤ 1
  • E. All real numbers

Based on our analysis, we can see that:

  • Option A is incorrect because the range of y = -5sin(x) is bounded between -5 and 5, not all real numbers.
  • Option B, -5 ≤ y ≤ 5, is the correct range we derived from our understanding of vertical stretches and reflections.
  • Option C is incorrect because it gives the range as -5/2 ≤ y ≤ 5/2, which corresponds to a vertical stretch by a factor of 5/2, not 5.
  • Option D is incorrect because it gives the range as -1 ≤ y ≤ 1, which is the range of the basic sine function y = sin(x), not y = -5sin(x).
  • Option E is incorrect because the range of y = -5sin(x) is bounded between -5 and 5, not all real numbers.

Therefore, the correct answer is B. The range of the function y = -5sin(x) is -5 ≤ y ≤ 5. This is because the vertical stretch by a factor of 5 expands the range, and the reflection across the x-axis does not change the range but inverts the function's behavior.

In this article, we have thoroughly examined how to determine the range of the function y = -5sin(x). By understanding the properties of the basic sine function, y = sin(x), and the effects of transformations such as vertical stretches and reflections, we were able to accurately identify the range as -5 ≤ y ≤ 5. The range of a function is a fundamental concept in mathematics, especially in trigonometry, and mastering it is essential for solving a variety of problems.

Key Takeaways:

  • The range of the basic sine function, y = sin(x), is -1 ≤ y ≤ 1.
  • Vertical stretches change the amplitude and, consequently, the range of the sine function.
  • Reflections across the x-axis invert the function but do not change the range.
  • For y = -5sin(x), the vertical stretch by a factor of 5 expands the range to -5 ≤ y ≤ 5, and the reflection does not alter the range.

By breaking down the function into its components and understanding the effect of each transformation, we can confidently determine the range of trigonometric functions. This approach can be applied to a wide variety of similar problems, making it a valuable skill in mathematics.

Understanding the range of trigonometric functions is not just an abstract mathematical concept; it has practical applications in various fields such as physics, engineering, and computer science. For example, in physics, the sine function is used to model oscillatory motion, and understanding its range helps in predicting the maximum displacement of an object. In engineering, trigonometric functions are used in signal processing and circuit analysis, and knowing their ranges is crucial for designing systems that operate within certain limits. Therefore, a solid grasp of these concepts is essential for students pursuing careers in STEM fields.