Problem 105 - Precalculus

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: Understand Reflection About the Y-Axis

When a function is reflected about the y-axis, the transformation replaces every instance of \(x\) in the function with \(-x\). This changes the direction of the graph horizontally.

Step 2: Apply the Reflection Transformation

Given the function \(f(x) = 4^x\), reflect it about the y-axis by replacing \(x\) with \(-x\). So, the new function becomes \(f(x) = 4^{-x}\).

Key Concepts

Graph Transformations

Graph transformations are like changing the way a picture looks without altering the content. Think of it as resizing, moving, or flipping the image of a graph. These transformations can change the position, orientation, and even the shape of the graph on a coordinate plane. Two common types are translations, where you move the graph up, down, left, or right, and reflections, where you flip the graph.

In our exercise, the concept of reflection is crucial. By applying transformations to the graph of a function, you can better understand its behavior and how different elements of the graph interact with each other. Whether it's stretching it wider or making it taller, transformations help you see the function from a new angle.

• Vertical transformations involve shifting the graph up or down.
• Horizontal transformations move the graph left or right.
• Reflections flip the graph across a specific axis.
• Scaling changes the size of the graph differently in x- or y-directions.

Reflection Across the Y-Axis

Reflection across the y-axis, in simple terms, is like looking at yourself in a mirror. Everything you see on your left becomes right in the mirror. For graphs, this means that you take every point on the graph and flip it to the opposite side of the y-axis.

Mathematically, if you have a function like our example, such as \(f(x) = 4^x\), and you want to reflect it across the y-axis, you replace every occurrence of \(x\) with \(-x\). This changes the graph of the function horizontally.

• Original point \((x, y)\) becomes \((-x, y)\) after reflection.
• The graph maintains its shape, just flipped over the y-axis.
• This transformation is useful to analyze symmetries in functions.

Function Notation

Function notation is a way of expressing the output of a function given an input. It's a simple, yet powerful tool to describe functions without getting into lengthy explanations each time. When you see \(f(x)\), it represents the function name with \(x\) as its variable or input.

This notation is not just a fancy symbol; it has practical uses. It allows for clear communication of what operation is being performed on the input to produce an output. When you perform transformations like reflections, function notation helps express these new forms easily.

• \(f(x)\) denotes the function with \(x\) as an input.
• After a transformation like reflection, \(f(x)\) can change to \(f(-x)\).
• This change makes it easier to understand how the function behaves with different operations or transformations.