Problem 36 - 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 \(x\) -axis

Step-by-Step Solution

Answer

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

Step 1: Identify the Transformation

The exercise requires us to reflect the function \(f(x) = 4^x\) about the \(x\)-axis. To reflect a graph about the \(x\)-axis, we multiply the entire function by \(-1\).

Step 2: Apply the Transformation

Apply the transformation by multiplying the function \(f(x) = 4^x\) by \(-1\). This gives us the transformed function \(g(x) = -4^x\).

Step 3: Verify the Transformation

Reflecting \(f(x) = 4^x\) about the \(x\)-axis results in flipping the graph over the \(x\)-axis. This can be checked by ensuring that if a point \((a, b)\) is on \(f(x)\), then \((a, -b)\) is on \(g(x)\).

Key Concepts

Graph Reflections

Reflections in graphing are like holding a mirror to your function. Effects occur based on which axis is being mirrored.
When reflecting a function about the x-axis, you flip the graph upside-down. To perform this transformation, multiply the entire function by -1.

• If you start with \( f(x) = 4^x \), the reflection about the x-axis transforms it into \( g(x) = -4^x \).
• This means every y-value of the original function will be negated in the new function.
• For instance, if a point on the graph of \( f(x) \) is \((a, b)\), it transforms to \((a, -b)\) on the graph of \( g(x) \).

Graph reflections help make visual symmetries clear and often offer mathematical insights into the behavior of different functions by comparing them to their reflections.

Algebraic Transformations

Algebraic transformations involve applying algebraic operations, like addition, subtraction, multiplication, or division, to modify functions. These transformations change the appearance of graphs in predictable ways.
Here's how a few common transformations affect functions:

• **Vertical Translations:** Add a constant to \( f(x) \) to move the graph up or subtract to move it down, \( f(x) + c \).
• **Horizontal Translations:** Modify the input \( x \) to shift the graph left or right. For example, \( f(x - c) \) moves it to the right.
• **Reflections:** Multiply by -1 to reflect over an axis—\( -f(x) \) means reflect over the x-axis.

Algebraic transformations are key in graphing functions because they let you visualize how basic functions change as we tweak them algebraically. They form the basis for understanding more complex movements of graphs.

Exponential Functions

Exponential functions are critical in mathematics for modeling rapid growth or decay. Their general form is \( f(x) = a^x \), where \( a \) is a constant.
Here's what to know about these functions:

• An exponential function like \( f(x) = 4^x \) grows fast because the variable \( x \) is in the exponent.
• The base \( a \), which is 4 in this case, dictates how quickly it grows. A base larger than 1 gives growth, while between 0 and 1 results in decay.
• Exponential functions have unique properties, such as always being positive and having a horizontal asymptote along the x-axis.

Understanding exponential functions is crucial because they appear in various real-world scenarios, from finance to biology, representing dynamics like compounding interest or population growth.