Problem 23 - College Algebra

Question

For the following exercises, graph the function and its reflection about the \(x\) -axis on the same axes. $$ f(x)=\frac{1}{2}(4)^{x} $$

Step-by-Step Solution

Answer

Graph the function and its reflection using steps of plotting key points and drawing smooth curves.

Step 1: Understand the Function

The given function is \( f(x) = \frac{1}{2}(4)^x \). This is an exponential function where the base is 4, and it is multiplied by \( \frac{1}{2} \). The variable \( x \) is in the exponent, indicating that this is an increasing exponential function.

Step 2: Graph the Original Function

To graph \( f(x) = \frac{1}{2}(4)^x \), calculate and plot several points. For example: \( f(0) = \frac{1}{2}(4)^0 = \frac{1}{2} \), \( f(1) = \frac{1}{2}(4)^1 = 2 \), and \( f(-1) = \frac{1}{2}(4)^{-1} = \frac{1}{8} \). Plot these points and sketch a smooth curve through them to represent the function.

Step 3: Determine the Reflection

The reflection of the function about the \( x \)-axis can be found by changing the sign of the function outputs. This reflects each point downward across the \( x \)-axis, giving the function \( g(x) = -\frac{1}{2}(4)^x \).

Step 4: Graph the Reflected Function

Using the reflection function \( g(x) = -\frac{1}{2}(4)^x \), find and plot the corresponding reflected points. For example: \( g(0) = -\frac{1}{2} \), \( g(1) = -2 \), and \( g(-1) = -\frac{1}{8} \). Draw a smooth curve through these points for the reflected graph.

Step 5: Finalize the Graph

On the same set of axes, draw the graphs of \( f(x) = \frac{1}{2}(4)^x \) and its reflection \( g(x) = -\frac{1}{2}(4)^x \). The original function should curve upwards as \( x \) increases, while the reflection curves downwards.

Key Concepts

Reflection of Graphs

When learning about exponential functions, observing their reflections can be quite insightful. Reflection in mathematical terms means flipping a graph over a specific axis. In our case, we are reflecting over the x-axis. By reflecting an exponential graph, you're essentially flipping its direction.
Imagine the graph of an exponential function climbing upwards. Its reflection on the x-axis will have it curve downwards instead. This means if you begin with a point on the original function, simply invert the y-value to find its reflection.
For example:

• If a point on the original graph is (1, 2), its reflected point will be (1, -2).
• This results in a downward curve, mirroring the shape of the original function.

Reflection transformations are helpful when comparing two functions to understand differences in behavior.

Exponential Growth

Exponential growth is a critical concept in mathematics, depicting a process that increases rapidly over time. In the function given, \(f(x) = \frac{1}{2}(4)^x\), we see an example of exponential growth.
Here’s how exponential growth manifests:

• The base number, here 4, dictates the nature of growth. The higher the base, the steeper the graph.
• Positive exponents result in values growing larger, producing an upward curve as x increases.
• Even small changes in x can lead to significant increases in f(x), illustrating rapid growth.

Exponential functions like this are particularly useful in modeling real-world phenomena such as population growth, where numbers escalate quickly.

Function Transformation

Understanding function transformations is key to mastering exponential functions. Transformations alter the graph's appearance without changing its essential structure.
There are various types of transformations:

• Vertical Shifts: Adding or subtracting a constant to the function moves the graph up or down.
• Horizontal Shifts: Adding or subtracting a constant inside the function (involving x) shifts it left or right.
• Reflections: As we discussed earlier, altering the sign of the function reflects it over the x-axis.

Consider the exercise's use of a reflecting transformation. By transforming the function from \( f(x) = \frac{1}{2}(4)^x \) to \( g(x) = -\frac{1}{2}(4)^x \), only the orientation changes, showcasing the power of transformations in graphing.