Problem 33 - College Algebra with Corequisite Support

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 \(f(x) \quad 3\) units downward

Step-by-Step Solution

Answer

The new function is \( g(x) = 4^x - 3 \).

Step 1: Understanding the Original Function

The original function given is \( f(x) = 4^x \). This function is an exponential function with base 4, meaning it grows rapidly as \( x \) increases. The graph of this function passes through the point (0, 1) because \( 4^0 = 1 \).

Step 2: Analyzing the Transformation

The transformation required is to shift the function \( f(x) \) 3 units downward. This means we need to adjust the output value of the function by subtracting 3 from it. Mathematically, this is done by subtracting 3 from the original function \( f(x) = 4^x \).

Step 3: Writing the Transformed Function

To write the new function after the transformation, we modify the original function \( f(x) = 4^x \) by subtracting 3, resulting in: \[ g(x) = 4^x - 3 \]. This function represents the graph of \( f(x) = 4^x \) shifted 3 units downward.

Key Concepts

Exponential Functions

Exponential functions are special mathematical functions where a constant base is raised to a variable exponent. In our example, the function is defined as \( f(x) = 4^x \). Here, the base is 4, and the exponent is \( x \). These functions are known for their rapid growth or decay.

• If the base is greater than 1, the function exhibits exponential growth. This means the function values shoot up very quickly as \( x \) increases.
• If the base is between 0 and 1, the function shows exponential decay, decreasing rapidly as \( x \) increases.

To observe these changes, it's crucial to understand how exponential functions graphically represent this behavior. These graphs typically have a horizontal asymptote, which in many cases is the x-axis or \( y = 0 \). As \( x \) approaches negative infinity, the function values become very small, nearing but never reaching zero.Exponential functions are fundamental in many fields, including biology for population growth, finance for compound interest, and physics for radioactive decay.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting a graph of a function. When applied to exponential functions, these transformations help us understand how changes in function equations affect their graphical appearance. In the given exercise, the transformation applied to the graph of \( f(x) = 4^x \) is a vertical shift. Specifically, we are shifting the graph 3 units downward. This transformation will move each point on the graph the same number of units in the vertical direction, thus maintaining the shape of the graph.The mathematical operation for this downward shift is simple:

• To move a graph \( k \) units downward, we subtract \( k \) from the function. This results in the new function \( g(x) = f(x) - k \).

For our exercise, by subtracting 3 from each value of \( f(x) \), we obtain \( g(x) = 4^x - 3 \), which shifts the entire graph of \( f(x) \) three units downwards. The horizontal asymptote, originally at \( y = 0 \), is now at \( y = -3 \). This indicates that as \( x \) becomes very negative, \( g(x) \) approaches but never quite reaches \( y = -3 \).

Function Notation

Function notation is a way to express the relationship between inputs and outputs in mathematics. It allows us to easily denote and manipulate functions. Normally, a function is written as \( f(x) \), where \( f \) denotes the function and \( x \) represents any input value from the function’s domain. This concept might seem simple, but it is crucial for understanding how to handle transformations.

• Any transformation on the function is incorporated into this notation. For instance, \( g(x) = 4^x - 3 \) shows both the base function and its transformation.
• Function notation helps us visually and analytically understand how the input values (x-values) are transformed into output values (function values).

This notation also clarifies instructions when performing operations on a function. For example, using \( g(x) = f(x) - 3 \) clearly shows the subtraction of 3 from each output of \( f(x) \), thus transforming the function. Being comfortable with function notation is essential as it serves as a foundation for more advanced mathematical topics, including calculus and differential equations.