Problem 37 - 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. Refl \(\operatorname{ct} f(x)\) about the \(y\) -axis

Step-by-Step Solution

Answer

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

Step 1: Understanding the Transformation

Reflecting a function about the y-axis involves creating a horizontal reflection. For the function \(f(x) = 4^x\), this means we need to replace \(x\) with \(-x\) in the function's equation.

Step 2: Apply the Reflection

To reflect the function \(f(x) = 4^x\) about the y-axis, we substitute \(x\) with \(-x\). Therefore, the reflected function is \(f(-x) = 4^{-x}\).

Step 3: Write the Transformed Function

The function resulting from reflecting \(f(x) = 4^x\) about the y-axis is \(f(x) = 4^{-x}\). This function represents the graph of \(4^x\) flipped horizontally.

Key Concepts

Graphing Exponential Functions

Exponential functions are powerful tools in mathematics. They allow us to model real-world phenomena like population growth and radioactive decay. The general form of an exponential function is \(f(x) = a^x\), where \(a\) is a positive constant and \(x\) is the exponent.
The graph of an exponential function is quite distinct. When \(0 < a < 1\), the graph decreases towards zero as \(x\) increases; it's called exponential decay. When \(a > 1\), like our function \(f(x) = 4^x\), the graph rises sharply, which is exponential growth.

Some key characteristics to remember when graphing exponential functions are:

• The graph passes through the point (0, 1) because any non-zero number raised to the power of 0 is 1.
• It never touches the x-axis; it gets closer and closer but doesn't actually meet, which is known as an asymptote.
• The graph is always increasing or always decreasing, based on whether \(a\) is less than or greater than 1.

Understanding the basic shape and behavior of these functions is crucial before applying transformations like reflections or translations.

Reflection Over Y-Axis

A reflection over the y-axis is a type of horizontal transformation. It flips the graph of a function horizontally across the y-axis. For our function \(f(x) = 4^x\), reflecting it over the y-axis results in substituting \(x\) with \(-x\) to create the transformed function \(f(-x) = 4^{-x}\).
This transformation changes the direction in which the graph rises or falls. For \(f(x) = 4^x\), which rapidly increases as \(x\) moves to the right, its reflection \(f(-x) = 4^{-x}\) will decrease as \(x\) increases, showcasing exponential decay.

This transformation is especially important because:

• It changes the rate of increase to a rate of decrease or vice versa.
• The function's behavior becomes mirrored with a line of symmetry along the y-axis.
• This technique is widely used in symmetry analysis and curve sketching.

Recognizing reflections helps in predicting and understanding the effects on the graph's shape and direction.

Horizontal Transformation

Horizontal transformations shift or stretch the graph of a function leveragely along the x-axis. Reflecting over the y-axis is a special case of horizontal transformation. Instead of simply shifting the graph, the image is flipped.
To understand a horizontal transformation, we start with the substitution \(x\) with another term, often \(x - h\). For example:

• Replacing \(x\) with \(x - 2\) moves the graph 2 units to the right.
• Replacing \(x\) with \(x + 3\) moves the graph 3 units to the left.

In the case of reflection over the y-axis, we substitute \(x\) with \(-x\). This doesn’t translate the graph, but reverses its direction. In our function \(f(x) = 4^x\), applying this results in \(f(-x) = 4^{-x}\), changing how the graph behaves as independent variable \(x\) increases.

Horizontal transformations act directly on the x-values, affecting:

• The graph's position along the x-axis.
• The symmetry and orientation when reflected.
• Visualization of inverse effects of horizontal scaling or translation.

By mastering these transformations, the study of functions and their graphs becomes easier to visualize and interpret.