Plotting the Graph of f(x) 2^-x: A Comprehensive Guide

In this article, we will explore the process of plotting the graph of the function f(x) 2^-x. We will start by understanding the parent function 2^x, and how it can be shifted and reflected to form the desired graph.

Parent Function: 2^x

The parent function of interest is 2^x. This function plays a crucial role in understanding the behavior of exponential functions. Here are some key characteristics:

Domain: All real numbers ((x in mathbb{R})) Range: All positive real numbers ((y > 0)) Y-intercept: (f(0) 2^0 1) X-intercept: None Increasing function for all (x) Asymptotic to the x-axis as (x rightarrow -infty)

To graph (2^x), we start by plotting some key points. For instance, at (x 0), (f(x) 1); and as (x) increases, the function grows exponentially.

Shifting the Function Up by 1 Unit

A common practice when working with exponential functions is to shift them vertically to avoid overlap. By shifting the parent function (2^x) up by 1 unit, we create a new function:

(2^x 1)

Domain: All real numbers ((x in mathbb{R})) Range: All real numbers greater than 1 ((y > 1)) Y-intercept: (f(0) 1) X-intercept: None Asymptotic to the line (y 1) as (x rightarrow -infty)

Graphing (2^x 1) involves plotting the same points as (2^x) but shifting them up by 1 unit. This serves as a reference point for further transformations.

Shifting and Reflecting to Form f(x) 2^-x

Now, we focus on the function f(x) 2^-x. This function is the reflection of (2^x) over the y-axis, followed by a horizontal reflection. Here is the step-by-step process:

Step 1: Reflecting (2^x) over the y-axis To reflect (2^x) over the y-axis, we replace (x) with (-x), giving us (2^-x). This means we take the original function and flip it horizontally over the y-axis. The domain remains the same, but the range changes due to the reflection.

Step 2: Reflecting (2^-x) over the x-axis To further transform it to (2^-x) from (2^-x), we need to reflect it over the x-axis. This is achieved by taking (-2^-x). However, since we are dealing with an exponential of a negative exponent, the reflection does not directly negate the function; it rather changes its behavior.

Plotting (2^-x), we know:

Domain: All real numbers ((x in mathbb{R})) Range: All positive real numbers ((y > 0)) Y-intercept: (f(0) 2^0 1) X-intercept: None Asymptotic to the x-axis as (x rightarrow infty) Decreasing function for all (x)

The function (2^-x) is an even function, which means it is symmetrical with respect to the y-axis. We can verify this by checking that (f(-x) f(x)).

Symmetry and Reflection

To ensure the function (2^-x) is accurately represented, we can draw the graph for (x geq 0) and then reflect it horizontally over the y-axis to complete the graph for all (x).

Drawing for (x geq 0) Starting at (x 0), (f(0) 1). As (x) increases, the function decreases exponentially towards 0.

Complete drawing for the whole x-axis For (x The graph is symmetrical with respect to the y-axis, as (f(-x) 2^{-(-x)} 2^x 2^{-x} f(x)).

In summary, the graph of (f(x) 2^-x) is a reflection of the original exponential function (2^x) over the y-axis and then over the x-axis. This results in a symmetric graph around the y-axis.

Conclusion

Understanding the process of plotting the graph of (f(x) 2^-x) requires a thorough understanding of exponential functions and their transformations. By following the steps of reflecting and translating, we can accurately graph this function and appreciate its symmetry and behavior.