Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Graph each function. Do not use a calculator. $$f(x)=3^{x}$$

Step-by-Step Solution

Answer

Graph the exponential curve with key points like \( (0, 1) \), \( (1, 3) \), and approaching zero as x decreases.

Step 1: Understand the Function

The function is an exponential function of the form \( f(x) = a^x \). In this case, \( a = 3 \), which tells us that the base of the exponential function is 3. Exponential functions have a characteristic shape called an exponential curve.

Step 2: Identify Key Points

Key points for exponential functions often include the y-intercept and any easy-to-calculate points. At \( x = 0 \), \( f(0) = 3^0 = 1 \), so the y-intercept is \( (0, 1) \). Other key points might be \( x = 1 \), where \( f(1) = 3^1 = 3 \), and \( x = -1 \), where \( f(-1) = 3^{-1} = \frac{1}{3} \).

Step 3: Determine the Behavior of the Function

As \( x \) increases, \( 3^x \) increases exponentially. Conversely, as \( x \) decreases, \( 3^x \) approaches 0 but never actually reaches it. This shows that the graph has a horizontal asymptote at \( y = 0 \).

Step 4: Plot Key Points on the Coordinate Plane

Draw the coordinate axes and plot the points calculated: \( (0, 1) \), \( (1, 3) \), and \( (-1, \frac{1}{3}) \). These points will help give the shape of the curve.

Step 5: Sketch the Graph

Using the key points and understanding of the behavior, draw the exponential curve. Start from very close to the x-axis on the negative side, pass through the points, and continue upwards as \( x \) becomes positive. Ensure the curve approaches the y-axis at \( y = 1 \) and rises steeply on the right.

Key Concepts

Exponential Growth

Exponential growth refers to how exponential functions like \( f(x) = 3^x \) increase at a rapid and accelerating rate. These functions have a base raised to a variable exponent, making them powerful in representing growth in various fields like biology, finance, and technology.
In our exercise, the exponential growth is evident by examining how quickly the function values increase as \( x \) becomes larger. For instance:

• At \( x = 0 \), the function value is \( f(0) = 1 \).
• At \( x = 1 \), it significantly jumps to \( f(1) = 3 \).
• At \( x = 2 \), it exponentially grows further to \( f(2) = 9 \).

As the exponent value increases, each increment results in larger and larger jumps, demonstrating the nature of exponential growth.

Horizontal Asymptote

A horizontal asymptote in the graph of an exponential function is a horizontal line that the curve approaches but never quite touches. For the function \( f(x) = 3^x \), the horizontal asymptote is the line \( y = 0 \).
This is because, as the value of \( x \) decreases towards negative infinity, \( f(x) = 3^x \) approaches zero but never actually becomes zero.

• When \( x = -1 \), the function value is \( f(-1) = \frac{1}{3} \).
• As \( x \) continues to decrease, more fractions smaller than \( \frac{1}{3} \) are produced.

No matter how small the value of \( x \) becomes, the function will never cross the line \( y = 0 \), which is a key characteristic of exponential decay behavior.

Key Points in Graphing

Identifying key points is essential to understand and sketch the graph of exponential functions like \( f(x) = 3^x \). These points serve as reference markers that help in drawing the curve accurately. There are typically a few points that are easy to calculate:

• **Y-intercept**: At \( x = 0 \), the function value is \( f(0) = 3^0 = 1 \), giving the y-intercept at the point \((0, 1)\).
• **Forward Point**: At \( x = 1 \), you have \( f(1) = 3^1 = 3 \), showing a forward increasing point at \((1, 3)\).
• **Backward Point**: At \( x = -1 \), the function is \( f(-1) = 3^{-1} = \frac{1}{3} \), indicating behavior as \( x \) goes negative, at the point \((-1, \frac{1}{3})\).

These key points are plotted and connected to visualize the exponential curve effectively.

Function Behavior Analysis

Analyzing the behavior of an exponential function like \( f(x) = 3^x \) involves observing its trend as the input values change. This helps to intuitively understand the nature and progression of the graph.
Here's a breakdown:

• **As \( x \) increases**: The function rises sharply, demonstrating exponential growth. The rate of increase becomes quicker with larger \( x \) values.
• **As \( x \) decreases**: The function decreases towards zero but never reaches it. This shows the curve's approach towards the horizontal asymptote \( y = 0 \).
• **Overall Shape**: The curve starts from the near-zero region and becomes steeper as it progresses.

This analysis reveals the core dynamics and gives insights into how changes in \( x \) influence \( f(x) \), making the function's graph predictable and clear.