Question

Sketch the graph of the exponential equation. $$y=0.5^{x}$$

Short answer

The graph of the function \(y = 0.5^x\) shows exponential decay. The plot passes through the points: (-2, 4), (-1, 2), (0, 1), (1, 0.5), and (2, 0.25).

Step-by-step solution

Understand the exponential function

An exponential function is a mathematical function of the form \(f(x) = a^x\), where 'a' is a positive real number other than 1. In this function, \(y = 0.5^x\), the base is 0.5, which is less than 1. This indicates that the graph will show exponential decay.

Determine key points

Choose convenient values for \(x\) to find corresponding \(y\) values. Some good choices could be \(x = -2, -1, 0, 1, 2\). Calculating these will give the key points to plot: (-2, 4), (-1, 2), (0, 1), (1, 0.5), and (2, 0.25).

Graph the function

Plot these points on the graph. Connect the points. The graph should trail off to the right, never quite reaching the x-axis. This is because \(y = 0.5^x\) is always positive. On the left, the graph will go upwards sharply. Approaching negative infinity, the \(y\)-values increase exponentially.