Graphing Transformations of Exponential Functions

In this guide, we will explore how to graph a series of transformed exponential functions, starting with the basic exponential function y e^x. We will discuss various transformations, including reflections, vertical and horizontal scaling, and shifts, to generate new functions.

Basics of Exponential Functions

The basic exponential function, y e^x, is a fundamental graph recognizable by its characteristic shape. As x increases, the function grows rapidly, and conversely, as x decreases, the function approaches zero but never actually reaches it. This graph is key to understanding more complex transformations.

Transforming y e^x

Reflection over the x-axis

To reflect the graph of y e^x over the x-axis, we change the sign of the function, resulting in y -e^x. This new graph is a mirror image of the original but below the x-axis.

Scaling and Shifting Adjustments

1. To scale all y values by half, we have y -0.5e^x. This compresses the graph vertically by a factor of two, making the function less steep. 2. To shift all y values by one, we get y -0.5e^x 1. This moves the graph upwards by one unit. 3. To scale the graph by a factor of two vertically and then shift it up by two, we have y 2 - e^x. This dilates the graph vertically by a factor of two and then shifts it upwards by two units.

Additional Transformations

Another example involves the function r(x) 1 - 0.5e^{-x}. This can be constructed as follows:

Reflection and Squeezing

To derive r(x), first reflect the graph of y e^x over the y-axis to get y e^{-x}. Next, compress that graph vertically by a factor of two to obtain y 0.5e^{-x}. Finally, reflect that graph over the line y 0.5 to get y 1 - 0.5e^{-x}.

Determining Maxima and Minima

Let's consider the function g(x) 2 - e^x. To find its maxima and minima:

As x goes to infinity, g(x) goes to negative infinity. e^{-x} is a monotonically decreasing function, so the minimum of g(x) is negative infinity. As x approaches negative infinity, g(x) approaches 2. Therefore, the maximum value of g(x) is 2. At x 0, g(0) 1. This is a critical point where the function reaches its maximum value of 2 as x approaches negative infinity.

Graphing Tips and Tools

To obtain a rough sketch of these functions without precise plotting, one can use a hand-drawn graph of y e^x and perform the described transformations. Alternatively, using a graphing calculator or a tool like Wolfram Alpha can provide a more accurate representation. For an interactive approach, Geogebra is a valuable tool.

Note: The content herein offers a comprehensive overview of exponential function transformations and includes practical examples and explanations.