Problem 23 - College Algebra

Question

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

Plot \( f(x) = \frac{1}{2}(4)^x \) and its reflection \( g(x) = -\frac{1}{2}(4)^x \) on the same graph.

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 the multiplier is \( \frac{1}{2} \). As \( x \) increases, \( f(x) \) grows exponentially.

Step 2: Calculate Key Points for the Function

Choose key values for \( x \) like -1, 0, 1, and 2 to find corresponding \( f(x) \) values. Compute:- \( f(-1) = \frac{1}{2} (4)^{-1} = \frac{1}{8} \)- \( f(0) = \frac{1}{2} (4)^{0} = \frac{1}{2} \)- \( f(1) = \frac{1}{2} (4)^{1} = 2 \)- \( f(2) = \frac{1}{2} (4)^{2} = 8 \)

Step 3: Graph the Function

Plot the points \((-1, \frac{1}{8}), (0, \frac{1}{2}), (1, 2), (2, 8)\) on a coordinate plane. Connect these points smoothly because the function is continuous, forming an upward-growing curve.

Step 4: Reflect the Function About the X-Axis

To find the reflection of \( f(x) \) over the x-axis, take \( -f(x) \). The reflected function is \( g(x) = -\frac{1}{2}(4)^x \). Calculate the reflection points using key \( x \) values:- \( g(-1) = -\frac{1}{8} \)- \( g(0) = -\frac{1}{2} \)- \( g(1) = -2 \)- \( g(2) = -8 \)

Step 5: Graph the Reflected Function

Plot the points corresponding to \( g(x) \) as \((-1, -\frac{1}{8}), (0, -\frac{1}{2}), (1, -2), (2, -8)\). Connect these points smoothly, creating a downward-growing curve. These points are a mirror image of their counterparts in \( f(x) \) along the x-axis.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They appear in the general form of \( f(x) = a \, b^x \) where \( a \) is a constant multiplier, and \( b \) is the base of the exponential. In our example, \( f(x) = \frac{1}{2} (4)^x \), the base is \( 4 \), meaning that for every one-unit increase in \( x \), the function's value quadruples.
These functions grow or decay exponentially. If the base \( b \) is greater than 1, the function grows rapidly as \( x \) increases. If \( b \) is between 0 and 1, it decays. Here, the base is 4, so \( f(x) \) increases sharply.

• Exponential growth implies a rapid increase in values as \( x \) increases.
• A constant multiplier, like \( \frac{1}{2} \) here, scales the function vertically.
• Understanding how exponential functions work is essential for graphing them accurately.

Graphing Functions

Graphing is the process of plotting a function's values on a coordinate plane. For exponential functions like \( f(x) = \frac{1}{2} (4)^x \), we plot several key points to understand the function's behavior. We start by selecting strategic \( x \) values, calculate corresponding \( f(x) \) values, and plot these coordinates.
The points calculated in our solution, \((-1, \frac{1}{8}), (0, \frac{1}{2}), (1, 2), (2, 8)\), provide a rough outline of the curve. Connecting these points smoothly helps visualize how the curve behaves between plotted points:

• It's crucial to choose both positive and negative \( x \) values for a complete view.
• Plotting helps identify the exponential curve's direction and steepness.
• A continuous curve suggests there's more than just these plotted points.

Coordinate Plane

The coordinate plane is a two-dimensional space formed by two perpendicular number lines, the x-axis, and y-axis. We use it to represent and visualize mathematical functions graphically. Every point on this plane is defined by an \( (x, y) \) pair, making it a powerful tool for interpreting equations and their reflections.
Understanding the axes is crucial. The x-axis is horizontal, while the y-axis is vertical. When plotting functions like \( f(x) \) and its reflection, each point is placed according to its \( x \) and \( f(x) \) values. This clear visual representation makes complex algebraic functions more comprehensible:

• Positive x-values are to the right, negative ones to the left of the y-axis.
• Positive y-values are above, while negative ones are below the x-axis.
• The origin, \( (0,0) \), is where these axes intersect.

Function Transformation

Function transformation involves changing a graph's position or shape without altering its essential properties. In our exercise, we see a simple transformation: reflection.
Reflecting a function like \( f(x) = \frac{1}{2}(4)^x \) over the x-axis amounts to multiplying its output by -1. This transformation yields \( g(x) = -\frac{1}{2}(4)^x \). The reflection changes the upward curve of \( f(x) \) to a downward one:

• The reflection makes the function curve a mirror image along the x-axis.
• This concept can apply to both function reflections and rotations.
• Understanding transformations is essential for manipulating and interpreting different function types.