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Chapter 13: Problem 24

Graph each exponential function. $$f(x)=-\left(\frac{1}{3}\right)^{x}$$

Short Answer

Expert verified
In short, the graph of the exponential function \(f(x) = -\left(\frac{1}{3}\right)^x\) is a decay function with a y-intercept at (0, -1) and a horizontal asymptote at y = 0. The negative sign causes the graph to be reflected over the x-axis. To graph the function, plot the y-intercept, draw the asymptote, then sketch the decay function below the x-axis as it approaches the asymptote.

Step by step solution

01

Identify the base of the exponential function

The function is given by \(f(x) = -\left(\frac{1}{3}\right)^x\). The base of the exponential function is \(\frac{1}{3}\). Since the base is between 0 and 1, we know that it is a decay function.
02

Determine the y-intercept

To find the y-intercept, we have to substitute x=0 into the function: \(f(0) = -\left(\frac{1}{3}\right)^{0}\) Any number raised to the power of 0 is 1, so we get: \(f(0) = -1\) Thus, the y-intercept is -1.
03

Determine the horizontal asymptote

For all exponential functions, the horizontal asymptote is the x-axis, or y = 0, because as x tends to infinity or negative infinity in an exponential function, the value of the function approaches 0.
04

Identify the effect of the negative sign

Since the function is multiplied by -1, it reflects the graph over the x-axis. Instead of decaying towards the x-axis from above, the graph will decay towards the x-axis from below.
05

Graph the exponential function

Now, let's graph the function using the information we gathered: 1. The base is \(\frac{1}{3}\), so the function decays. 2. The y-intercept is -1 (the point (0, -1) on the graph). 3. The horizontal asymptote is the x-axis (y = 0). 4. The negative sign reflects the graph over the x-axis. Begin by plotting the y-intercept at (0, -1). Next, draw the horizontal asymptote at y = 0. Since the graph is a decay function, it will start decreasing from the left and approaching the x-axis but never crossing it. The graph is reflected over the x-axis, so instead of being above the x-axis, it will be below it. By combining all of these properties, we can create the graph of the exponential function \(f(x) = -\left(\frac{1}{3}\right)^x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Exponential Functions
Graphing exponential functions can initially seem daunting, but with a few key pieces of information, it becomes an approachable task. To graph any exponential function, such as \(f(x) = -\left(\frac{1}{3}\right)^x\), we need to understand its properties and behavior.
  • Base: The base, \(\frac{1}{3}\), is crucial as it tells us about the growth (greater than 1) or decay (between 0 and 1) nature of the function.
  • Negative Sign: The negative sign in front of the function reflects the graph, flipping it over the x-axis, making the function decrease on the graph's lower side.
  • Y-intercept: This is where the graph crosses the y-axis, calculated by setting \(x\) to 0, giving us \(f(0) = -1\).
Break down each property individually to understand their roles, then piece them together. Begin plotting the y-intercept and horizontal asymptote. Sketch this by identifying decay reduction meaning the values will taper as \(x\) increases. Remember, exponentials like \(-\left(\frac{1}{3}\right)^x\) never actually reach zero鈥攖hey just creep infinitesimally close!
Understanding Decay Functions
Exponential decay functions, like \(f(x) = -\left(\frac{1}{3}\right)^x\), have their own distinctive characteristics. Unlike exponential growth, decay occurs when the base of the exponent is a fraction between 0 and 1. These functions model how quantities reduce over time. For our function, the decay factor is \(\frac{1}{3}\). This means as \(x\) increases by 1, the value of \(f(x)\) is multiplied by \(\frac{1}{3}\).
  • Graph Behavior: The magnitude of the function decreases rapidly initially and slows as it approaches zero.
  • Reflection and Decay: The negative sign in front causes the decay trend to reflect over the x-axis, flipping the negative values downward.
By putting these clues together, understand the decay pattern. As \(x\) increases, the graph falls towards negative infinity, subtly approaching the x-axis.
Horizontal Asymptote in Exponential Functions
Horizontal asymptotes are fascinating and relevant in the realm of exponential functions. They represent a "boundary" that the graph approaches but never quite reaches. For the function \(f(x) = -\left(\frac{1}{3}\right)^x\), the horizontal asymptote is particularly significant.
  • Definition: The horizontal asymptote is essentially a guideline that demonstrates where the function will settle in the long run. For many exponential functions, this boundary lies along the line \(y=0\) (the x-axis).
  • Function Behavior: As \(x\) tends to both positive and negative extremes, the values get closer to \(y=0\), though never crossing or touching that line.
In our specific function, the graph seeks this horizontal line from below, providing insight into the function's long-term behavior. Understanding this concept is crucial for complete comprehension of exponential graphs and their tendencies.

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