Problem 46 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

Sketch a smooth curve passing through (0,1) and increasing rapidly, approaching the x-axis as an asymptote.

Step 1: Understand the Function

The given function is an exponential function, written as \( f(x) = 4^x \). This type of function involves a base raised to the power of \( x \), where the base here is 4. The function has a characteristic shape, usually showing rapid growth or decay.

Step 2: Identify Key Characteristics

For the exponential function \( f(x) = 4^x \), note the following key characteristics:- The domain is all real numbers, \( x \in (-\infty, \infty) \).- The range is positive real numbers, \( y \in (0, \infty) \).- The y-intercept is at \( (0, 1) \), since \( 4^0 = 1 \).- The graph is an increasing function with no x-intercepts.

Step 3: Plot Key Points

Choose some easy points to calculate their \( y \)-values and plot them:- At \( x = 0 \), \( f(0) = 4^0 = 1 \), so plot \( (0, 1) \).- At \( x = 1 \), \( f(1) = 4^1 = 4 \), so plot \( (1, 4) \).- At \( x = -1 \), \( f(-1) = 4^{-1} = 0.25 \), plot \( (-1, 0.25) \).- Continue this pattern for additional points like \( x = 2 \) and \( x = -2 \).

Step 4: Draw the Graph

Using the points plotted in Step 3, draw a smooth curve starting from near zero on the left (asymptotically approaching the x-axis) and rising sharply as \( x \) increases. Make sure the curve passes through all the plotted points. The graph should steadily increase, reflecting the positive base of the exponential term.

Step 5: Sketch the Asymptote

Since \( f(x) = 4^x \) never reaches zero, the x-axis (\( y = 0 \)) acts as a horizontal asymptote. Indicate this by noting the tendency of the graph to approach the x-axis but never touch it or cross it.

Key Concepts

Graphing Techniques

When graphing exponential functions like \( f(x) = 4^x \), remember to follow a few key techniques to create an accurate representation. Start with plotting the easily calculated points that help form the framework of your graph.

• Begin by identifying the y-intercept, which for \( f(x) = 4^x \) is at \((0, 1)\), since any base raised to the power of zero equals one.
• Select additional x-values, such as -1, 1, and 2, to calculate corresponding y-values. These calculations will yield points like \((-1, 0.25)\), \((1, 4)\), and \((2, 16)\).
• Plot these points on your graph accurately. Ensure they form a smooth curve that appears to rise sharply as \(x\) increases.
• Remember the horizontal asymptote \( y = 0 \) as the curve approaches the x-axis but never crosses it.

By following these steps, you'll display the essence of exponential growth, creating a clear visual guide for the function's behavior.

Function Characteristics

Understanding the characteristics of exponential functions like \( f(x) = 4^x \) is key to mastering their behavior. These functions have specific traits that set them apart from other function types.

• The **domain** is all real numbers, \( x \in (-\infty, \infty) \), indicating that you can input any real number and receive a valid output.
• The **range** is \( y \in (0, \infty) \), meaning the output is always positive. This reflects the fact that powers of any positive number (like 4 in this case) are always positive.
• There is no x-intercept, as the graph never touches the x-axis. Since 4 raised to any real number doesn't equal zero, the function never hits y=0.
• The y-intercept is at \((0, 1)\), since \( 4^0 = 1 \). This point is crucial as it anchors the curve on the graph.

These characteristics are common in exponential functions and understanding them helps predict and explain their steep growth or decay.

Exponential Growth

Exponential functions like \( f(x) = 4^x \) demonstrate exponential growth, a pattern that becomes quickly apparent when plotting these functions.

• The base of the function, 4 in this case, is greater than one, signaling that the function increases as x increases. This forms a distinctly steep curve upward.
• This type of growth is faster than linear growth. Even small changes in x lead to large changes in y. For example, from \( x=1 \) to \( x=2 \), there is a jump from y=4 to y=16.
• Exponential growth is often used to model real-world scenarios where quantities double rapidly, such as populations, financial investments, or the spread of a virus.

Recognizing exponential growth patterns not only aids in understanding graphs but also in grasping how the function might model real-world phenomena effectively.