Problem 32 - College Algebra

Question

For the following exercises, start with the graph of \(f(x)=4^{x}\). Then write a function that results from the given transformation. Shift \(f(x) 4\) units upward

Step-by-Step Solution

Answer

The transformed function is \( 4^x + 4 \).

Step 1: Understand the Original Function

The original function is given as \( f(x) = 4^x \). It represents an exponential function with base 4.

Step 2: Identify the Transformation

We need to shift the function \( f(x) \) 4 units upward. This is a vertical shift and affects the function's output.

Step 3: Apply the Transformation

To shift a function \( f(x) \) 4 units upward, you add 4 to the function. Hence, the transformed function becomes \( f(x) + 4 = 4^x + 4 \).

Key Concepts

Understanding Exponential Functions

Exponential functions are a fascinating class of functions that grow or decay at an ever-increasing rate. The primary identifying feature of an exponential function is its constant base raised to a variable exponent, such as in the equation \( f(x) = 4^x \). This function means that the base number 4 is raised to the power of \( x \), where \( x \) can be any real number.

Exponential functions are commonly found in real-world applications like population growth, radioactive decay, and interest calculations in finance. These functions are characterized by a rapid increase (or decrease) depending on whether the base is greater or less than one. In our example, since the base is 4, which is greater than one, \( f(x) = 4^x \) is a model of exponential growth.

• The graph of an exponential function with a base greater than one always increases and approaches zero as \( x \) approaches negative infinity.
• Exponential growth results in a curve that gets steeper and rises quickly.

This basic understanding sets the foundation for further transformations, such as vertical shifts.

Exploring Vertical Shifts

Vertical shifts are a type of transformation that adjusts a graph up or down along the y-axis without altering its shape. This is achieved by adding or subtracting a constant to the function. When a function experiences a vertical shift, every point on the graph moves uniformly up or down.

In the context of the given problem, we are shifting the graph of \( f(x) = 4^x \) upward by 4 units. To do this, we need to add 4 to the output of the function, resulting in the transformed function \( f(x) + 4 = 4^x + 4 \).

• The graph of \( 4^x \) moves parallel to its original position.
• The increase in the y-value means the entire graph is raised without changing its curvature.
• This does not affect the rate of growth or the function's base, only its vertical position.

Vertical shifts are particularly useful in modeling situations where there is an added or reduced baseline level, such as adjustments for initial quantities in a practical scenario.

Graphing Functions and Transformations

Graphing functions helps visualize the changes and transformations applied to them. It's a crucial part of understanding how different transformations, like translations and shifts, affect the function's graph.

When you graph the original exponential function \( f(x) = 4^x \), you'll notice it starts low and rises quickly as \( x \) increases. If you apply the vertical shift and graph the function \( 4^x + 4 \), you will see that the entire curve is shifted 4 units higher on the y-axis.

• The x-intercept remains the same for such vertical transformations.
• The y-intercept shifts upward by the amount of the shift, which in this case is 4 units.
• The overall shape and steepness of the graph do not change.

By graphing both the original and transformed functions, one can clearly observe the vertical shift. It's a powerful visual tool that helps solidify the understanding of transformations in functions.