Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Graph each function. Do not use a calculator. $$f(x)=\left(\frac{1}{4}\right)^{x}$$

Step-by-Step Solution

Answer

The function \( f(x) = \left(\frac{1}{4}\right)^{x} \) is a decreasing exponential graph, crossing the y-axis at (0, 1) and approaching the x-axis as \( x \) increases.

Step 1: Determine the Base of the Exponential Function

In the given function, \( f(x) = \left(\frac{1}{4}\right)^x \), the base is \( \frac{1}{4} \). This is important because it tells us how the function behaves. Since the base is between 0 and 1, the function will be decreasing; as \( x \) increases, \( f(x) \) decreases.

Step 2: Understand the Key Features

For exponential functions of the form \( a^x \), where \( 0 < a < 1 \), the graph passes through the point (0, 1) because any number raised to the power of 0 is 1. As \( x \to \infty \), \( f(x) \to 0 \), and as \( x \to -\infty \), \( f(x) \to \infty \). The function never actually touches the x-axis, so the x-axis is a horizontal asymptote.

Step 3: Calculate Key Points

Evaluate the function at some key values to help sketch the graph. For instance, - At \( x = 0 \), \( f(0) = \left( \frac{1}{4} \right)^0 = 1 \).- At \( x = 1 \), \( f(1) = \left( \frac{1}{4} \right)^1 = \frac{1}{4} \).- At \( x = 2 \), \( f(2) = \left( \frac{1}{4} \right)^2 = \frac{1}{16} \).- At \( x = -1 \), \( f(-1) = \left( \frac{1}{4} \right)^{-1} = 4 \).

Step 4: Plot the Points

Using the calculated values, plot the points (0, 1), (1, \( \frac{1}{4} \)), (2, \( \frac{1}{16} \)), and (-1, 4) on a coordinate plane. These points will guide the shape of the graph as it decreases from left to right.

Step 5: Draw the Graph

Connect the plotted points with a smooth curve. Ensure the curve approaches the x-axis as \( x \to \infty \), reflecting the horizontal asymptote. The curve should rise steeply as \( x \to -\infty \). This gives the overall shape of the decreasing exponential graph.

Key Concepts

Graphing

Graphing exponential functions can be an exciting journey if you understand the basic properties that define their shape. The given function is a classic example of an exponential one:

• The base determines the behavior of the graph and whether it increases or decreases.
• The intercept and direction are crucial for sketching without calculators.

To start graphing, identify key points by choosing a few values of \( x \). This helps you plot the curve smoothly. For \( f(x) = \left( \frac{1}{4} \right)^x \), these points are:

• \( x = 0 \Rightarrow f(0) = 1 \)
• \( x = 1 \Rightarrow f(1) = \frac{1}{4} \)
• \( x = 2 \Rightarrow f(2) = \frac{1}{16} \)
• \( x = -1 \Rightarrow f(-1) = 4 \)

With these coordinates, start plotting them on your graph. Use a smooth curve to connect these dots. Two things to watch for:

• Approach the x-axis as \( x \to \infty \).
• Curve upwards steeply as \( x \to -\infty \).

This creates the characteristic shape of an exponential decay.

Base of Exponential Function

The base in an exponential function like \( f(x) = a^x \) is pivotal because it dictates how the function behaves. Here, \( a = \frac{1}{4} \). This number tells us that the graph will decrease because:

• The base is between 0 and 1.
• Each additional unit increase in \( x \) results in a smaller fraction for \( f(x) \).

For a base greater than 1, the function would increase instead. Understanding this is key to knowing the general direction of the graph without calculation. With a base like \( \frac{1}{4} \), as \( x \) increases, the answer for \( f(x) \) gets smaller and approaches zero, but never quite reaches it – an important attribute of decreasing exponential functions.

Horizontal Asymptote

A horizontal asymptote in exponential functions serves as an invisible boundary that the graph approaches but never crosses. For the function \( f(x) = \left(\frac{1}{4}\right)^x \), the x-axis itself is the horizontal asymptote.This occurs because:

• As \( x \to \infty \), \( f(x) \to 0 \).
• The function values keep getting smaller but will never be zero.

This behavior is fundamental for recognizing the limits of an exponential function. By knowing how to identify and interpret the horizontal asymptote, you can sketch the graph more accurately and appreciate how it behaves over different values of \( x \). In this example, no matter how far to the right you go along the x-axis, \( f(x) \) will not touch or dip below the axis, showcasing how the concept of asymptotes helps shape the function's personality.