Problem 45 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=3^{-x} $$

Step-by-Step Solution

Answer

The graph of \( f(x) = 3^{-x} \) is a decreasing exponential curve with a horizontal asymptote at \( y = 0 \).

Step 1: Understanding the Function

The given function is an exponential function of the form \( f(x) = 3^{-x} \). This means that the function decreases as \( x \) increases because the base, 3, is raised to the power of \( -x \). This is similar to exponential decay.

Step 2: Determine Key Points

Select key values of \( x \) and compute the corresponding \( f(x) \) values. For instance, calculate \( f(x) \) at \( x = -2, -1, 0, 1, 2 \).- \( x = -2 \): \( f(-2) = 3^{2} = 9 \)- \( x = -1 \): \( f(-1) = 3^{1} = 3 \)- \( x = 0 \): \( f(0) = 3^{0} = 1 \)- \( x = 1 \): \( f(1) = 3^{-1} = \frac{1}{3} \approx 0.33 \)- \( x = 2 \): \( f(2) = 3^{-2} = \frac{1}{9} \approx 0.11 \)

Step 3: Plot the Points

On a coordinate plane, plot the points from \( (-2, 9), (-1, 3), (0, 1), (1, 0.33), (2, 0.11) \). These points illustrate the behavior of the function as \( x \) changes.

Step 4: Draw the Graph

Connect the plotted points with a smooth curve to represent the graph of \( f(x) = 3^{-x} \). Note the shape of the curve: it starts high on the left and decreases, approaching zero as \( x \) increases.

Step 5: Analyze the Graph

The graph of \( f(x) = 3^{-x} \) is a decreasing exponential function. As \( x \) moves from negative to positive, \( f(x) \) approaches zero but never actually reaches it, indicating a horizontal asymptote at \( y = 0 \).

Key Concepts

Graphing Exponential Functions

Graphing exponential functions can initially seem daunting, but breaking it down into simple steps can help. An exponential function, like \( f(x) = 3^{-x} \), typically has a pattern of growth or decay depending on the exponent. In this case, we have a negative exponent, which signals decay. To graph it, start by choosing a few key values for \( x \). Compute the corresponding \( f(x) \) values, for instance, for \( x = -2, -1, 0, 1, 2 \), and plot these points on a coordinate plane.

Once you've plotted these points, they should illustrate the characteristic exponential curve. For \( f(x) = 3^{-x} \), the points will create a curve that starts high on the left and moves downward to almost touch the horizontal axis as \( x \) increases. Connecting the dots with a smooth line will reveal the complete graph of the function.

• Choose key points and compute \( f(x) \)
• Plot points on the coordinate plane
• Draw a smooth curve through the points

This process gives you a visual representation of how exponential functions behave under different conditions.

Exponential Decay

Exponential decay occurs when the function value decreases as the input, \( x \), increases. In the function \( f(x) = 3^{-x} \), the negative exponent causes the base of 3 to be raised to negative powers. This results in smaller and smaller values as \( x \) becomes larger. Unlike exponential growth, where values increase rapidly, decay results in a gradual decrease. This function type often describes processes like radioactive decay or cooling, where quantities diminish over time.

The decrease follows a specific pattern: for every one unit increase in \( x \), the value of \( f(x) \) is scaled down by a consistent factor, in this example, by a factor of 3, due to the base. This predictable decreasing pattern means the graph never rises again once it falls. It's a concept that underpins many scientific and real-world processes, making it incredibly useful in various applications.

• Function decreases as \( x \) increases
• Graph falls but never rises
• Common in natural processes

Horizontal Asymptote

In the graph of an exponential decay function like \( f(x) = 3^{-x} \), you'll notice that as \( x \) gets larger, the curve seems to approach a particular value without ever actually touching it. This happens because of the horizontal asymptote. For this function, the horizontal asymptote is at \( y = 0 \). This means that as \( x \) moves toward infinity, \( f(x) \) gets closer and closer to zero but never actually becomes zero.

Understanding the concept of a horizontal asymptote is essential for graphing exponential functions accurately. It helps to anticipate how the curve will behave at extreme values of \( x \). You can think of this line like a boundary that the function approaches but never crosses. This invisible barrier gives clear insight into the long-term behavior of the graph and is crucial in predicting future outcomes in numerous practical scenarios.

• Function approaches but never reaches zero
• Acts as a boundary line for the curve
• Helps predict the graph's behavior at extreme \( x \) values