Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Graph each function. Do not use a calculator. $$f(x)=4^{x}$$

Step-by-Step Solution

Answer

The function \( f(x) = 4^x \) is an increasing exponential curve, crossing the y-axis at (0,1) and passing through (1,4) and (-1, \( \frac{1}{4} \)).

Step 1: Understand Basic Behavior

The function \( f(x) = 4^x \) is an exponential function. It will take the form of a curve that increases rapidly as \( x \) becomes larger. For \( x < 0 \), the function approaches zero but never touches the x-axis, as it can never be negative.

Step 2: Identify Key Points

To sketch the graph, identify key points of the function. The y-intercept, where \( x = 0 \), is \( f(0) = 4^0 = 1 \). As \( x = 1 \), the function value is \( f(1) = 4^1 = 4 \). Similarly, for \( x = -1 \), we have \( f(-1) = 4^{-1} = \frac{1}{4} \).

Step 3: Sketch the Graph

Plot the identified key points: (0,1), (1,4), and (-1, \( \frac{1}{4} \)). Connect these points with a smooth curve, remembering that the curve should get steeper as \( x \) increases and should approach the x-axis as \( x \) becomes more negative without touching it.

Key Concepts

Graphing Exponential Functions

Graphing an exponential function, like \( f(x) = 4^x \), involves plotting various points to illustrate how the function behaves across different values of \( x \). In an exponential function, the base (in this case, 4) determines the rate of growth. To start graphing, plot the points derived from evaluations at specific \( x \) values, such as \( x = 0, 1, \) and \( -1 \). These key values help define the shape of the curve.

• For \( x = 0 \), the function becomes \( 4^0 = 1 \), which establishes a crucial y-intercept at (0,1).
• With \( x = 1 \), the point is (1,4) since \( 4^1 = 4 \).
• For \( x = -1 \), the value is \( \frac{1}{4} \), leading to the point (-1, \( \frac{1}{4} \)).

Break the graph into segments. From your point at (0,1), move steadily upwards towards (1,4) and onwards, showing the rapid increase typical of exponential growth. As \( x \) decreases into negative numbers, the curve approaches the x-axis, illustrating its consistent behavior without crossing.

Key Points in Exponential Graphs

To accurately graph an exponential function, identifying key points is essential. These points are found by substituting simple values of \( x \) into the function. By focusing on these calculations:

• Y-intercept: This is typically the easiest key point to find. For \( f(x) = 4^x \), it's where \( x = 0 \), resulting in \( 4^0 = 1 \).
• Positive growth point: Setting \( x = 1 \) gives us a point at (1, 4) because \( 4^1 = 4 \).
• Near-zero behavior: When \( x = -1 \), \( 4^{-1} \) equals \( \frac{1}{4} \), a small but important detail showing the graph's behavior as it decreases.

Use these critical points to distribute the curve correctly on the coordinate plane. They ensure that you capture the essence of the exponential function's behavior, particularly in its rapid rise and approach to the axis.

Understanding Asymptotic Behavior

Asymptotic behavior refers to the characteristic of a graph as it approaches a line, called an asymptote, without ever quite touching it. In the particular case of \( f(x) = 4^x \), the graph nears the x-axis as \( x \) approaches negative infinity. Yet, it never actually crosses or reaches it.Key concepts related to asymptotic behavior:

• Horizontal Asymptote: For this exponential function, y = 0 (the x-axis) is the horizontal asymptote. This means that no matter how small \( x \) becomes, \( f(x) \) will always be positive, never zero.
• Exponential Growth: From right to left, as \( x \) becomes more negative, \( f(x) \) rapidly decreases, illustrating the close approach to y = 0.

Understanding this behavior clarifies why the graph trends sharply towards the axis, yet always remains above it. It's a hallmark of exponential functions, showcasing their unique tendency to approach but not quite meet their horizontal asymptote.