Problem 6 - Algebra A Combined Function

Question

Graph each exponential function. $$ y=\left(\frac{1}{5}\right)^{x} $$

Step-by-Step Solution

Answer

The graph of \( y = \left( \frac{1}{5} \right)^{x} \) is a decreasing exponential curve with a horizontal asymptote at \( y = 0 \).

Step 1: Understand the Function

We need to graph the function \( y = \left( \frac{1}{5} \right)^{x} \). This is an exponential function where the base of the exponent is \( \frac{1}{5} \). When the base of an exponential function is between 0 and 1, such as \( \frac{1}{5} \), the function is decreasing.

Step 2: Create a Table of Values

Select a few values of \( x \) to calculate corresponding \( y \) values. For example, calculate \( y \) for \( x = -2, -1, 0, 1, 2 \). - For \( x = -2 \): \( y = \left( \frac{1}{5} \right)^{-2} = 5^2 = 25 \) - For \( x = -1 \): \( y = \left( \frac{1}{5} \right)^{-1} = 5 \) - For \( x = 0 \): \( y = \left( \frac{1}{5} \right)^{0} = 1 \) - For \( x = 1 \): \( y = \left( \frac{1}{5} \right)^{1} = \frac{1}{5} \) - For \( x = 2 \): \( y = \left( \frac{1}{5} \right)^{2} = \frac{1}{25} \) This gives us the pairs \((x, y): \,( -2, 25), (-1, 5), (0, 1), (1, \frac{1}{5}), (2, \frac{1}{25}) \).

Step 3: Plot the Points

On graph paper or a digital graphing tool, plot the points obtained from the table: \((-2, 25)\), \((-1, 5)\), \((0, 1)\), \((1, \frac{1}{5})\), \((2, \frac{1}{25})\).

Step 4: Draw the Exponential Curve

Once the points are plotted, draw a smooth curve through the points. This curve should decrease from left to right, getting closer to the x-axis as \( x \) becomes larger, but never actually touching it. This is called an asymptote, specifically the horizontal asymptote at \( y = 0 \).

Step 5: Analyze the Graph

The graph of \( y = \left( \frac{1}{5} \right)^{x} \) shows exponential decay. It passes through the point \((0, 1)\) since any number to the power 0 is 1, and it declines as \( x \) increases, due to the base being a fraction.

Key Concepts

Graphing Exponential Functions

When graphing exponential functions, you're working with equations where variables are in the exponent, for example, \( y = \left(\frac{1}{5}\right)^{x} \). The graph's shape depends largely on the base. If the base is greater than 1, the function exhibits exponential growth, whereas if the base is between 0 and 1, it shows exponential decay.

Here's how to graph an exponential function:

• Identify the function's base. In our function, \( \left(\frac{1}{5}\right)^{x} \), the base is \( \frac{1}{5} \), indicating decay.
• Create a table of values by selecting \( x \) values and calculating the corresponding \( y \) values.
• Plot these points on a coordinate grid.
• Draw a smooth curve passing through the points. The curve will illustrate the overall behavior of the function.

Graphing gives you a visual representation of how the function behaves, showing increases or decreases, sometimes rapidly so. It's a vital tool in understanding the dynamics of exponential functions.

Exponential Decay

Exponential decay occurs in functions where the base is between 0 and 1, as it is in \( y = \left(\frac{1}{5}\right)^{x} \). This results in values getting smaller as the variable, \( x \), increases. Essentially, with each increase in \( x \), \( y \) shrinks by the function's factor.

Common examples of exponential decay include:

• Radioactive decay, where substances decay over time.
• Population decreases, perhaps due to a limited food supply.
• Any process where the rate of change decreases proportionally over time.

Understanding exponential decay helps in predicting long-term trends. In graphs, you'll see the curve flatten out as it approaches zero, but it never quite reaches it, illustrating its continuous, albeit slower, decline.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches. For our function, \( y = \left(\frac{1}{5}\right)^{x} \), the horizontal asymptote is at \( y = 0 \). This is because as \( x \) becomes very large, \( y \) approaches zero without actually reaching it.

Characteristics of horizontal asymptotes:

• They indicate potential limiting behavior of a function as \( x \) heads towards infinity or negative infinity.
• In exponential decay, like our example, the graph tails off and remains above the x-axis, depicting this behavior.
• Helps in understanding end behavior of functions and can be crucial in fields like calculus.

Recognizing horizontal asymptotes is key for correct graph interpretation, as they mark the threshold that the output values of a function will not exceed.

Table of Values

Creating a table of values is one of the first steps in graphing functions, such as the exponential function \( y = \left(\frac{1}{5}\right)^{x} \). It involves selecting specific \( x \) values, calculating the corresponding \( y \) values, and then using these pairs to plot the graph.

Steps for constructing a useful table:

• Choose a range of \( x \) values, typically around 5 to begin with, like -2 to 2.
• Calculate \( y \) for each \( x \). For example, \( x = -2 \) gives \( y = 25 \), whereas \( x = 2 \) gives \( y = \frac{1}{25} \).
• List these \( (x, y) \) pairs in a simple table format for quick reference and plotting.

This process aids in understanding the function's behavior and provides a foundational basis for graphing, allowing for visual verification of analytical insights.