Problem 52 - College Algebra with Modeling and Visualization

Question

Sketch a graph of \(y=f(x)\) $$ f(x)=\left(\frac{1}{4}\right)^{x} $$

Step-by-Step Solution

Answer

The graph is a decreasing exponential curve with a horizontal asymptote at \( y = 0 \).

Step 1: Understand the Function

The function given is \( f(x) = \left(\frac{1}{4}\right)^x \). This is an exponential function where the base is less than 1 (i.e., \( \frac{1}{4} \)). This suggests that as \( x \) increases, \( f(x) \) decreases, resulting in a decreasing exponential graph.

Step 2: Determine Key Points

Choose some values of \( x \) to find corresponding \( y \) values: - When \( x = 0, f(0) = \left(\frac{1}{4}\right)^0 = 1 \)- When \( x = 1, f(1) = \left(\frac{1}{4}\right)^1 = \frac{1}{4} \)- When \( x = 2, f(2) = \left(\frac{1}{4}\right)^2 = \frac{1}{16} \)- When \( x = -1, f(-1) = \left(\frac{1}{4}\right)^{-1} = 4 \)

Step 3: Plot the Points

Plot the points on a coordinate plane based on the values from Step 2: - \( (0, 1) \)- \( (1, \frac{1}{4}) \)- \( (2, \frac{1}{16}) \)- \( (-1, 4) \)

Step 4: Draw the Graph

Connect the plotted points with a smooth curve to form the graph of \( y = f(x) = \left(\frac{1}{4}\right)^x \). The curve should approach the x-axis as \( x \) increases (showing the exponential decay) and rise sharply as \( x \) decreases to negative values.

Step 5: Analyze Graph Behavior

Notice that the graph has a horizontal asymptote at \( y = 0 \), as \( f(x) \) approaches zero when \( x \) increases. The graph never touches the x-axis and continues to rise steeply for negative \( x \).

Key Concepts

Graphing Exponential Functions

Graphing exponential functions is all about understanding how the shape of the graph changes depending on the base of the exponential expression. An exponential function has the form \( f(x) = a^x \), where \( a \) is a constant. This particular function \( f(x) = \left(\frac{1}{4}\right)^x \) is a classic example of an exponential function with a base less than 1.
This means the graph will have a distinctive downward trend, known as exponential decay.
The steps to graph an exponential function are straightforward:

• Select a range of \( x \) values to substitute into the function and calculate corresponding \( y \) values.
• Plot these \( (x, y) \) points on a coordinate grid.
• Connect the points smoothly to observe the characteristic curve of the exponential function.

In the case of \( f(x) = \left(\frac{1}{4}\right)^x \), notice the angle and steepness of the graph, especially how fast it decreases as \( x \) increases, moving from left to right.

Exponential Decay

Exponential decay occurs when the base of an exponential function is between 0 and 1, such as \( \frac{1}{4} \) in the function \( f(x) = \left(\frac{1}{4}\right)^x \). This results in the function's values decreasing rapidly as \( x \) increases.
In simpler terms, with every increase in \( x \), the output \( f(x) \) gets smaller by a factor determined by the base - in this case, the base \( \frac{1}{4} \).
To see this in action, consider:

• When \( x = 0 \), \( f(0) = 1 \).
• As \( x +1 \), \( f(1) = \frac{1}{4} \), a much smaller value.
• Let \( x = 2 \), \( f(2) = \frac{1}{16} \), decreasing even further.

The concept of exponential decay is vital in fields like biology to model populations, or in finance for calculating depreciation of assets. It shows how quickly values can diminish under specific patterns or rules.

Asymptotes

An asymptote is a line that a graph approaches but never actually touches or crosses. In the context of exponential functions, horizontal asymptotes are prevalent.
For the function \( f(x) = \left(\frac{1}{4}\right)^x \), the graph approaches the horizontal asymptote \( y = 0 \) as \( x \) increases.
This indicates that the output \( f(x) \) gets closer and closer to zero, but never quite reaches it. The concept of asymptotes helps us understand the limits of functions and predict behavior even if a function seems to continue decreasing indefinitely.

• An asymptote provides a boundary of sorts for the graph to get infinitely close to.
• No matter how large \( x \) becomes, \( f(x) \) moves closer to 0 without hitting the x-axis.
• This idea is crucial when analyzing behavior of functions on graphs.

Asymptotes are a handy tool for understanding graphs beyond just the plotted points and help in identifying the ultimate trends in exponential functions.