Question

Graph the exponential function. $$y=\left(\frac{1}{3}\right)^{x}$$

Short answer

The graph of the function \(y=(\frac{1}{3})^{x}\) starts at the point (0,1) and decreases as it moves to the right, getting closer to the x-axis but never touching or crossing it, and increases as it moves to the left.

Step-by-step solution

Identify the Properties of The Exponential Function

The function given is \(y=(\frac{1}{3})^{x}\). Because the base, \(\frac{1}{3}\), is less than one, the function will decrease as \(x\) increases. The function will never be negative since it's an exponential function, and the y-intercept is at \(y=1\) when \(x=0\)

Graph the Function

The next step is to draw the graph. Start at the y-intercept (0,1) which is the point where the graph crosses the y-axis. As the x-values get larger, the graph gets closer to the x-axis but never touches or crosses the x-axis, showing a continual decrease. As x-values get smaller (negative), the graph increases. Thus the graph of the function in the xy-plane will be a decreasing curve.

Key concepts

Exponential Decay

Exponential decay describes a process where quantities decrease rapidly at a rate proportional to their current value. This phenomenon is prevalent in various areas like radioactive decay, depreciation of assets, or cooling of a hot object in a cooler environment.

In the context of exponential functions, decay happens when the base of the power (in this case, \(\frac{1}{3}\)) is between zero and one. Graphically, exponential decay is represented by a curve that approaches the x-axis as x increases, yet never touches or crosses it. This characteristic signifies that the value of y is shrinking, but theoretically will never reach zero.

The graph of the function \(y=(\frac{1}{3})^x\) will show a smooth curve that starts from the y-intercept and moves closer to the x-axis, but maintains a positive value irrespective of how large x becomes. It's crucial for students to understand that the y-values will decrease as the x-values increase, echoing the 'expiring' nature of decay in the graph.

Y-Intercept of a Graph

The y-intercept of a graph is a fundamental feature which indicates where a function or line crosses the y-axis of a Cartesian coordinate system. Specifically, it is the point where the x-value is zero.

In the given function \(y=(\frac{1}{3})^x\), finding the y-intercept is straightforward: you substitute x with zero. By doing so, like in our example, any number raised to the power of zero, according to the rules of exponents, is always one. Consequently, the y-intercept for our function is at the point (0,1).

Identifying the y-intercept is crucial for graphing because it provides an anchor point from which to plot other points of the curve. The y-intercept is particularly useful as a starting point when graphing by hand or predicting the behavior of the graph without creating a detailed plot.

Properties of Exponential Functions

Exponential functions have distinct properties that set them apart from other types of functions. One property of exponential functions is that they change by a factor of the base when the variable is increased by 1. This means that if you have a base of \(\frac{1}{3}\), every time x increases by 1, the value of y is multiplied by \(\frac{1}{3}\).

Another important property is that exponential functions are always positive. Since the base is positive and any real number raised to any power remains positive, the graph of an exponential function will never dip below the x-axis.

Closely related to this is the asymptotic behavior that the functions exhibit: as x approaches infinity, the function approaches zero but never actually reaches it, creating a horizontal asymptote at y=0. This is a critical concept because it impacts how the graph behaves at extreme values of x.

Understanding the properties of exponential functions helps students to craft the function's graph accurately and anticipate its behavior across different values of x.