Question

Graph the exponential function. $$y=\left(\frac{1}{5}\right)^{x}$$

Short answer

After successfully identifying the nature of the function, calculating the function for some suitable values and plotting those points, the graph of the function \(y=(\frac{1}{5})^{x}\) is obtained as a rapidly decreasing curve that gets closer and closer to the x-axis as x increases. It starts from y = 25 (when x=-2) and it goes downwards till it almost touches x-axis (but never does). It intersects y-axis at point (0,1).

Step-by-step solution

Understand the function

The function \(y=(\frac{1}{5})^{x}\) is an exponential function where the base is less than 1. For exponential functions where the base is less than 1, as x increases, y decreases rapidly. The overall shape would be a rapidly decreasing curve that gets closer and closer to the x-axis but never touches it, starting from the upper right and moving toward the bottom left, cutting the y-axis at y = 1.

Table of values

Just like other functions, plotting an exponential function \(y=(\frac{1}{5})^{x}\) can benefit from creating a table of values i.e., different ordered pairs (x, y) to identify key points on the graph. Some useful points to calculate might be for x=-2, -1, 0, 1, and 2. Here are these calculated: (-2, 25), (-1, 5), (0, 1), (1, 0.2), (2, 0.04).

Plot the function

Now, on the graph, plot these points and draw the curve such that it passes through all these points. The graph starts from the top, decreases rapidly as it moves to the right, and gets very close to the x-axis without touching or crossing it. The y-intercept of the curve is at y=1.