Problem 23 - College Algebra with Corequisite Support

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 both the original function and its reflection by plotting calculated points.

Step 1: Understand the Function

The function given is an exponential function: \( f(x) = \frac{1}{2}(4)^x \). This form represents a basic exponential growth where the base of the exponent is greater than 1, indicating growth as \( x \) increases.

Step 2: Find Key Points for the Function

Identify key points to plot on the graph. Some useful points to consider are those when \( x = -1, 0, 1, 2 \). Calculate these values: \( f(-1) = \frac{1}{2}(4)^{-1} = \frac{1}{8} \), \( f(0) = \frac{1}{2}(4)^0 = \frac{1}{2} \), \( f(1) = \frac{1}{2}(4)^1 = 2 \), and \( f(2) = \frac{1}{2}(4)^2 = 8 \).

Step 3: Plot the Original Function

Using the points calculated, plot \( f(x) \) on a graph. At \( x = -1 \), \( y = \frac{1}{8} \); at \( x = 0 \), \( y = \frac{1}{2} \); at \( x = 1 \), \( y = 2 \); and at \( x = 2 \), \( y = 8 \). Connect these points smoothly, as exponential functions have a smooth curve.

Step 4: Reflect the Function Across the x-axis

To find the reflection of the function about the \( x \)-axis, consider the new function \( g(x) = -f(x) = -\frac{1}{2}(4)^x \). This reflects each corresponding point to negative \( y \) values: if \( f(x) = y \), then \( g(x) = -y \).

Step 5: Find Key Points for the Reflection

Take the original key points and reflect them across the \( x \)-axis: If \( (x, y) \) was a point in the original function, its reflection will be \( (x, -y) \). So: when \( x = -1, g(-1) = -\frac{1}{8} \); when \( x = 0, g(0) = -\frac{1}{2} \); when \( x = 1, g(1) = -2 \); and when \( x = 2, g(2) = -8 \).

Step 6: Plot the Reflected Function

Plot the points for the reflected function on the same graph: \( (x, -y) \) values calculated in the previous step. At \( x = -1 \), \( y = -\frac{1}{8} \); at \( x = 0 \), \( y = -\frac{1}{2} \); at \( x = 1 \), \( y = -2 \); at \( x = 2 \), \( y = -8 \). Connect these points smoothly.

Key Concepts

Function Reflection

In mathematics, a function reflection is a transformation that "flips" the graph of a function across a specific line, such as the x-axis or y-axis. For exponential functions, reflecting across the x-axis is a common transformation. This involves taking the output values (y-values) of the function and changing their signs.
For instance, given the exponential function \( f(x) = \frac{1}{2}(4)^x \), its reflection across the x-axis is given by \( g(x) = -f(x) = -\frac{1}{2}(4)^x \).
This creates a new function whose graph is a mirror image of the original function across the x-axis.

• If the original function point is \((x, y)\), the reflected function's point will be \((x, -y)\).
• This alters the graph's appearance from above the x-axis to below it (or vice versa).

Understanding reflection is crucial for gaining a deeper insight into graph transformations and analyzing the impacts of multiplying the function's output by a negative number.

Graphing Techniques

Graphing techniques are essential tools for visualizing and analyzing functions. When dealing with exponential functions, understanding these techniques helps in sketching accurate representations of equations. For the function \( f(x) = \frac{1}{2}(4)^x \), graphing involves several key points:

• The function's base (in this case, 4) determines the growth rate. Since it's greater than 1, the function is exponential growth.
• The coefficient \( \frac{1}{2} \) affects the vertical stretch or compression.
• These functions are continuous and smooth, meaning they curve rather than having sharp angles.

Effective graphing involves:

• Choosing appropriate x-values (such as -1, 0, 1, 2) to calculate corresponding y-values.
• Marking these points on a graph with clarity and accuracy for correct graph curvature.
• Using smooth curves to connect the dots, reflecting how the function's output grows or shrinks.

These techniques will provide a reliable foundation for visualizing exponential functions and gearing up for further transformations or reflections.

Plotting Key Points

Plotting key points is a fundamental aspect of graphing functions, particularly for exponential equations. These points act as guideposts, helping you draw the function's overall shape accurately. Let's break it down for the function \( f(x) = \frac{1}{2}(4)^x \):

• **Starting Point:** Calculate \( f(0) = \frac{1}{2}(4)^0 = \frac{1}{2} \). This serves as a good reference point on the y-axis.
• **Proceed with Increments:** For practical plotting, compute values at easily manageable increments, such as -1, 1, and 2: \( f(-1) = \frac{1}{8} \), \( f(1) = 2 \), \( f(2) = 8 \).
• **Reflected Points:** Remember, for the function reflection, simply take each key y-value and flip its sign. This calculates reflected y-values aligning with key x-values.

By plotting these accurately, you lay down the backbone of your graph. Subtly connect the dots, ensuring the curve mirrors the expected "exponential curve" shaped by growth or decay dynamics.
Precision in plotting these key points allows for a clear visualization of how each x-value affects the function's outcome, aiding in mastering exponential function graphing and transformations.