Graphing Y=3 * 7^(-x) + 2 Exponential Function With Transformations

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In the realm of mathematics, understanding function transformations is crucial for visualizing and analyzing graphs efficiently. Exponential functions, with their characteristic rapid growth or decay, are a particularly interesting class of functions to explore through transformations. In this comprehensive guide, we will delve into the process of graphing the function y = 3 * 7^(-x) + 2 by meticulously applying a series of transformations to a basic exponential function. This step-by-step approach will not only provide you with the means to accurately plot the graph but also equip you with a deeper understanding of how different transformations affect the shape and position of exponential curves.

1. Identifying the Parent Function: The Foundation of Our Graph

To begin our journey into graphing y = 3 * 7^(-x) + 2, we must first identify the parent function. The parent function serves as the foundation upon which all transformations will be applied. In this case, the parent function is the basic exponential function y = 7^x. This function represents exponential growth, where the value of y increases rapidly as x increases. Understanding the shape and behavior of this parent function is essential for visualizing the impact of subsequent transformations.

The parent function y = 7^x has several key characteristics that are important to note:

  • Horizontal Asymptote: The graph approaches the x-axis (y = 0) as x approaches negative infinity. This line, y = 0, is the horizontal asymptote of the function.
  • Y-intercept: The graph intersects the y-axis at the point (0, 1). This is because any number raised to the power of 0 equals 1 (7^0 = 1).
  • Exponential Growth: As x increases, the value of y increases rapidly, resulting in a characteristic exponential growth curve.
  • Domain: The domain of the function is all real numbers, meaning that x can take any value.
  • Range: The range of the function is y > 0, meaning that the value of y is always positive.

By keeping these characteristics in mind, we can better understand how the transformations we apply will alter the graph of the parent function.

2. Reflection about the Y-axis: Unveiling the Impact of the Negative Exponent

The first transformation we encounter in the function y = 3 * 7^(-x) + 2 is the negative sign in the exponent. This negative sign indicates a reflection about the y-axis. In other words, the graph of y = 7^(-x) is a mirror image of the graph of y = 7^x across the y-axis. This transformation effectively reverses the direction of the exponential growth, turning it into exponential decay.

To visualize this reflection, consider the following:

  • Points on the Parent Function: If the point (a, b) lies on the graph of y = 7^x, then the point (-a, b) will lie on the graph of y = 7^(-x). For example, if (1, 7) is on the graph of y = 7^x, then (-1, 7) will be on the graph of y = 7^(-x).
  • Direction of Growth: The original exponential growth of y = 7^x, where y increased as x increased, is now reversed. In y = 7^(-x), y decreases as x increases, representing exponential decay.
  • Horizontal Asymptote: The horizontal asymptote remains at y = 0, as reflection about the y-axis does not affect the horizontal asymptote.
  • Y-intercept: The y-intercept remains at (0, 1), as reflection about the y-axis does not change the point where the graph intersects the y-axis.

Understanding the reflection about the y-axis is crucial for accurately graphing the transformed function. It sets the stage for the next transformations we will apply.

3. Vertical Stretch: Amplifying the Function's Magnitude

Next, we encounter the coefficient 3 in front of the exponential term in the function y = 3 * 7^(-x) + 2. This coefficient represents a vertical stretch by a factor of 3. A vertical stretch multiplies the y-coordinate of each point on the graph by the given factor, in this case, 3. This transformation effectively stretches the graph vertically away from the x-axis.

To understand the effect of the vertical stretch, consider the following:

  • Y-coordinates: If the point (a, b) lies on the graph of y = 7^(-x), then the point (a, 3b) will lie on the graph of y = 3 * 7^(-x). For example, if (-1, 7) is on the graph of y = 7^(-x), then (-1, 21) will be on the graph of y = 3 * 7^(-x).
  • Vertical Distance from the Asymptote: The vertical stretch increases the distance of the graph from the horizontal asymptote. Points that were close to the asymptote are now further away.
  • Y-intercept: The y-intercept is also affected by the vertical stretch. The original y-intercept of (0, 1) is transformed to (0, 3) because 3 * 1 = 3.
  • Horizontal Asymptote: The horizontal asymptote remains at y = 0, as vertical stretches do not affect horizontal asymptotes.

The vertical stretch significantly alters the shape of the graph, making it steeper and further away from the x-axis. This transformation prepares the graph for the final step: the vertical translation.

4. Vertical Translation: Shifting the Graph Upward

The final transformation in the function y = 3 * 7^(-x) + 2 is the addition of 2. This represents a vertical translation upward by 2 units. A vertical translation shifts the entire graph vertically, either upward or downward, without changing its shape or orientation. In this case, adding 2 shifts the graph upward by 2 units.

To understand the effect of the vertical translation, consider the following:

  • Y-coordinates: If the point (a, b) lies on the graph of y = 3 * 7^(-x), then the point (a, b + 2) will lie on the graph of y = 3 * 7^(-x) + 2. For example, if (-1, 21) is on the graph of y = 3 * 7^(-x), then (-1, 23) will be on the graph of y = 3 * 7^(-x) + 2.
  • Horizontal Asymptote: The horizontal asymptote is also affected by the vertical translation. The original horizontal asymptote of y = 0 is shifted upward by 2 units to become y = 2.
  • Y-intercept: The y-intercept is shifted upward as well. The original y-intercept of (0, 3) is transformed to (0, 5) because 3 + 2 = 5.
  • Range: The range of the function is also affected by the vertical translation. The original range of y > 0 is shifted upward to become y > 2.

The vertical translation completes the transformation process, giving us the final graph of y = 3 * 7^(-x) + 2. The graph is an exponential decay curve that has been reflected about the y-axis, stretched vertically by a factor of 3, and translated upward by 2 units. The horizontal asymptote is now at y = 2, and the y-intercept is at (0, 5).

5. Putting it all Together: The Final Graph and Key Characteristics

By carefully applying each transformation step-by-step, we have successfully graphed the function y = 3 * 7^(-x) + 2. The final graph exhibits the following key characteristics:

  • Exponential Decay: The graph shows exponential decay, meaning that the value of y decreases as x increases.
  • Reflection about the Y-axis: The graph is a reflection of the basic exponential function about the y-axis.
  • Vertical Stretch: The graph has been stretched vertically by a factor of 3, making it steeper than the parent function.
  • Vertical Translation: The graph has been translated upward by 2 units, shifting the entire graph and its asymptote upward.
  • Horizontal Asymptote: The horizontal asymptote is at y = 2.
  • Y-intercept: The y-intercept is at (0, 5).
  • Domain: The domain of the function is all real numbers.
  • Range: The range of the function is y > 2.

By understanding the individual effects of each transformation, we can accurately predict the shape and position of the final graph. This step-by-step approach is a powerful tool for graphing a wide variety of transformed functions.

Conclusion: Mastering Transformations for Exponential Functions

Graphing exponential functions through transformations is a fundamental skill in mathematics. By systematically applying transformations such as reflections, stretches, and translations, we can accurately visualize and analyze complex functions like y = 3 * 7^(-x) + 2. This process not only enhances our understanding of exponential functions but also provides a framework for analyzing other types of functions and their transformations. Mastering these techniques unlocks a deeper understanding of the relationship between equations and their graphical representations, empowering you to tackle a wider range of mathematical problems with confidence.

Remember, the key to success in graphing transformed functions lies in identifying the parent function and carefully applying each transformation step-by-step. With practice and a solid understanding of the underlying principles, you can confidently graph even the most complex transformed functions. Embrace the power of transformations and unlock the beauty and elegance of mathematical graphs.