Question

Sketch the graph of the function. $$ y=\left(\frac{1}{3}\right)^{x} $$

Short answer

The graph of function \( y=(\frac{1}{3})^{x} \) is a decreasing curve that gets closer to the x-axis as x goes to infinity but never touches it and rises to infinity as x goes to negative infinity. The function has a y-intercept at (0,1) and no x-intercept.

Step-by-step solution

Identify the Function

The function \( y=(\frac{1}{3})^{x} \) is an exponential function because it has the form of \( y=b^{x} \) where b is a positive constant, and x is the exponent.

Determine the Key Points

To plot the function, we need some key points. Let's find the y-intercept by setting x=0. For any real number a we have \( a^0 = 1 \), so when \( x = 0 \), \( y = (\frac{1}{3})^{0} = 1 \). Therefore, the y-intercept of the function is at the point (0,1). For the x-intercept, the exponential function does not touch or cross the x-axis, so there is no x-intercept.

Sketch the Graph

First, plot the y-intercept at the point (0,1). Second, note that as x increases, the value of y decreases, but it never reaches 0. This gives the curve a decreasing shape. Also, because there is no x-intercept, the graph never crosses the x-axis. It keeps getting closer to the x-axis but never touches it. Hence, the graph of the function is a decreasing curve that approaches the x-axis as an asymptote as x goes to infinity, and it rises to infinity as x goes to negative infinity.

Key concepts

Sketching Exponential Functions

When sketching exponential functions, certain steps and observations guide us to an accurate representation. Starting with the basic form, exponential functions appear as y=b^x, where b is a positive real number not equal to 1.

For our example, y=(1/3)^x, the process involves identifying key points that help in shaping the graph. The most crucial of these is the y-intercept, which can be found by setting x to zero. Since any number to the power of zero is 1, the function will necessarily intercept the y-axis at the point (0,1).

As there is no x-intercept for exponential functions (they never touch or cross the x-axis), we observe how the function behaves as x increases or decreases from this point. Since 1/3 is less than 1, as x increases, the value of y gets smaller, revealing a decreasing pattern. It is essential to plot several points by choosing different x values to understand the curvature of the function. Moreover, arrows can be added to the ends of the curve to indicate the function extends infinitely in the direction of decreasing x and approaches an asymptote as x increases.

Exponential Function Properties

Exponential functions exhibit distinguishing properties that set them apart from other types of functions. An understanding of these properties is essential in accurately sketching their graphs.

One of the fundamental properties of exponential functions is that they are never zero or negative; the output values (y) will always be positive for a base b > 0. This is why our function, y=(1/3)^x, does not have an x-intercept and why the graph always remains above the x-axis.

Another noteworthy property is that the rate of growth or decay is proportional to the value of the function. For a base b > 1, the function exhibits exponential growth, where the graph rises sharply as x increases. Conversely, for 0 < b < 1, as in our example, the function shows exponential decay; the graph falls off as x increases but at an ever-decreasing rate.

These functions are also continuous and smooth, meaning there are no breaks, holes, or sharp turns in their graphs. Together, these properties yield a curve that either rises or falls exponentially, smoothly transitioning through all points plotted on the graph.

Asymptote of Exponential Functions

An asymptote is a line that a graph approaches but never quite reaches, no matter how far it extends along the x-axis. For the function y=(1/3)^x, the x-axis itself acts as a horizontal asymptote.

This behavior reflects a vital characteristic of exponential functions: as x becomes very large or very small, the value of y approaches a constant value, but does not equal it at finite x. In our example, as x goes to positive infinity, y gets infinitesimally close to zero without actually reaching zero — this is why the curve appears to flatten as it moves further along the positive x-axis.

Understanding the concept of an asymptote is crucial, as it affects how we sketch exponential functions, ensuring we do not mistakenly draw the graph as touching or crossing the asymptote. Remembering that the curve approaches but does not touch the x-axis sharpens our depiction of the exponential decay shown in this function.