Problem 37 - 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. Reflect \(f(x)\) about the \(y\) -axis

Step-by-Step Solution

Answer

The transformed function is \(f(x) = 4^{-x}\).

Step 1: Identify the Transformation

We need to reflect the graph of the function \(f(x) = 4^x\) about the \(y\)-axis. Reflecting a function about the \(y\)-axis affects the \(x\)-variable inside the exponent. This transformation involves negating the \(x\) in the function.

Step 2: Apply the Transformation to the Function

To reflect \(f(x) = 4^x\) about the \(y\)-axis, replace \(x\) with \(-x\) in the function. This gives us the transformed function \(f(-x) = 4^{-x}\).

Step 3: Simplify the Transformed Function

Express the function in a more standard form if necessary. In this case, \(4^{-x}\) is a simplified form already. However, we can also express it as \(\frac{1}{4^x}\) if needed.

Key Concepts

Reflecting Functions

Reflecting functions is a fundamental transformation in mathematics. Reflection involves flipping a graph over a specific line, and it's a key concept in altering the appearance of function graphs.
When we specifically reflect a function over the y-axis, it means we are changing the direction of the function along the x-axis.
To achieve this, we simply replace each instance of the x-variable in the function with -x. This changes the input of the function to its negative counterpart, effectively mirroring the graph across the y-axis.- Start with the function: Consider the given function, such as the exponential function \( f(x) = 4^x \).- Apply reflection: Substitute \( x \) with \(-x\), transforming it to \( f(-x) = 4^{-x} \).- Understand the effect: The graph will now appear as if you flipped it horizontally, which is especially evident for non-symmetric graphs.

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. They take the form \( f(x) = a^x \), where \( a \) is a positive real number known as the base.These functions are powerful in modeling growth and decay processes.
In the function \( f(x) = 4^x \), 4 is the base, which indicates fast growth as \( x \) increases.
A notable property of exponential functions is that they never touch the x-axis; they asymptotically approach it.Some key aspects include:

• Continuous growth or decay: Demonstrated by non-negative bases greater than 1 resulting in growth, whereas those between 0 and 1 lead to decay.
• Unbounded behavior: As x tends to infinity, \( a^x \) grows without bound, assuming \( a > 1 \).
• Simplicity of transformation: Exponential functions can easily be transformed through operations like reflections, shifts, and stretches.

Graph Transformations

Graph transformations encompass a wide range of modifications we can make to the graph of a function. These transformations can include shifts, stretches, reflections, and more. They allow us to visualize how changes in an equation affect the graph.In our specific exercise, reflecting the function \( f(x) = 4^x \) resulted in \( f(-x) = 4^{-x} \).
This acts as a horizontal flip, altering the direction of growth.Common graph transformations include:

• Vertical Shifts: Moving the graph up or down without changing its shape.
• Horizontal Shifts: Moving the graph left or right, essential for translating functions.
• Vertical and Horizontal Stretches: Scaling the graph in one direction to change its steepness or spread.
• Reflections: Flipping the graph across a line, such as the x-axis or y-axis.

Understanding these transformations helps in predicting the behavior and position of function graphs following any given transformation.