Graph y= 2^x and x = 2^y in the same rectangular coordinate system.

Verified step by step guidance

1. Identify the two functions to be graphed: the first is an exponential function \(y = 2^{x}\), and the second is \(x = 2^{y}\), which can be rewritten to express \(y\) in terms of \(x\) for easier graphing.
2. Rewrite the second equation \(x = 2^{y}\) by taking the logarithm base 2 of both sides to solve for \(y\): \(y = \log_{2}(x)\). This shows that the second graph is the logarithmic function, the inverse of the first.
3. Plot key points for \(y = 2^{x}\) by choosing several values of \(x\) (such as \(-2, -1, 0, 1, 2\)) and calculating the corresponding \(y\) values using \(y = 2^{x}\).
4. Plot key points for \(y = \log_{2}(x)\) by choosing several positive values of \(x\) (such as \(\frac{1}{4}, \frac{1}{2}, 1, 2, 4\)) and calculating the corresponding \(y\) values using \(y = \log_{2}(x)\).
5. Draw smooth curves through the plotted points for both functions on the same coordinate system, noting that the graphs are reflections of each other across the line \(y = x\) because they are inverse functions.

Key Concepts

Exponential Functions

An exponential function has the form y = a^x, where the base a is a positive constant not equal to 1. It models rapid growth or decay, and its graph passes through (0,1) with a smooth curve increasing or decreasing depending on the base. Understanding y = 2^x helps in plotting one of the given functions.

Inverse Functions and Their Graphs

The function x = 2^y can be seen as the inverse of y = 2^x by swapping variables. Inverse functions reflect across the line y = x, so graphing both on the same axes shows this symmetry. Recognizing this relationship aids in understanding the shape and position of both graphs.

Coordinate System and Plotting Points

Plotting functions requires understanding the rectangular coordinate system, where each point is defined by (x, y). To graph y = 2^x and x = 2^y, select values for x or y, compute corresponding points, and plot them accurately. This visual representation helps compare and analyze the functions.