Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Graph each function. Do not use a calculator. $$f(x)=\left(\frac{1}{3}\right)^{x}$$

Step-by-Step Solution

Answer

The function \( f(x) = \left( \frac{1}{3} \right)^x \) is an exponential decay graph that decreases from +∞ to 0.

Step 1: Analyze the Function Type

The given function \( f(x) = \left( \frac{1}{3} \right)^x \) is an exponential function. It has the form \( a^x \) where \( a = \frac{1}{3} \). Since the base \( \frac{1}{3} \) is between 0 and 1, this function represents exponential decay.

Step 2: Identify Key Characteristics

For the function \( f(x) = \left( \frac{1}{3} \right)^x \), the y-intercept is at \( x = 0 \), resulting in \( f(0) = 1 \). There is no x-intercept as the function never crosses the x-axis. As \( x \to +\infty \), \( f(x) \to 0 \), and as \( x \to -\infty \), \( f(x) \to +\infty \). Additionally, the function is always positive and decreasing.

Step 3: Plot Points for the Function

To graph the function, calculate a few key points:- \( f(-2) = \left( \frac{1}{3} \right)^{-2} = 9 \)- \( f(-1) = \left( \frac{1}{3} \right)^{-1} = 3 \)- \( f(0) = \left( \frac{1}{3} \right)^{0} = 1 \)- \( f(1) = \left( \frac{1}{3} \right)^{1} = \frac{1}{3} \)- \( f(2) = \left( \frac{1}{3} \right)^{2} = \frac{1}{9} \)

Step 4: Plot the Graph Using Key Points

Using the points from the previous step, plot them on a coordinate plane. Connect the points with a smooth curve that decreases from left to right, approaching the x-axis as \( x \to +\infty \), but never touching it. This illustrates the exponential decay behavior of the function.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions involves understanding the unique characteristics that set them apart from other functions. When you have an exponential function like \( f(x) = \left( \frac{1}{3} \right)^x \), it means you must graph a function where the base is raised to the power of \( x \). For this specific function, the base \( \frac{1}{3} \) is less than 1, which signals exponential decay.

To graph such a function without a calculator, it is essential to calculate a few values by substituting different \( x \) values into the function. For instance, you might find that \( f(-2) = 9 \), \( f(-1) = 3 \), \( f(0) = 1 \), \( f(1) = \frac{1}{3} \), and \( f(2) = \frac{1}{9} \).

These points help in creating a curve on the graph. When connected, they create a smooth line that moves from the upper left to the lower right, steadily getting closer to the x-axis but never crossing it. This graphical representation makes it visually clear how the function behaves.

Exponential Decay

Exponential decay is a concept that reflects how quantities decrease rapidly at first and then level off slowly. When a function like \( f(x) = \left( \frac{1}{3} \right)^x \) is examined, it is noted for being a type of exponential decay because its base, \( \frac{1}{3} \), is a fraction between 0 and 1.

In practical terms, exponential decay describes processes like radioactive decay, where the quantity decreases over time. For our function, as \( x \) increases, the value of \( f(x) \) rapidly shrinks towards zero, never reaching negative or zero itself.

This behavior is why we see the graph approach but never touch the x-axis. The decay illustrates a smooth, curved decline that showcases how effective such functions are in representing real-world scenarios where decline happens at a continuously decreasing rate.

Characteristics of Exponential Graphs

The characteristics of exponential graphs distinguish them from linear or quadratic graphs. With exponential functions, you initially notice the rapid change, either growth or decay. Here, \( f(x) = \left( \frac{1}{3} \right)^x \) presents exponential decay.

Some key characteristics to recognize include:

• **Y-Intercept**: At \( x = 0 \), \( f(x) = 1 \). This is where the function intersects the y-axis.
• **Asymptotic Behavior**: As \( x \to +\infty \), \( f(x) \to 0 \) — the graph approaches the x-axis but never touches it.
• **Monotonicity**: The function constantly decreases, showing that it's strictly declining as \( x \) gets larger.
• **Positivity**: The function remains positive; it does not cross into negative values.

These graphs provide a profound way to predict and understand behaviors that change exponentially over time. When graphing exponential functions, each unique characteristic offers insights into how the aspect of growth or decay functions in various circumstances.