Problem 91 - Precalculus

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 across the \(x\)-axis to visualize their symmetry.

Step 1: Understanding the Function

The function given is \( f(x) = \frac{1}{2}(4)^x \). This is an exponential function with a base greater than 1. The function will grow rapidly as \( x \) increases.

Step 2: Create a Table of Values for \(f(x)\)

Choose several \( x \)-values such as -2, -1, 0, 1, and 2, and calculate the corresponding \( f(x) \) values.- For \( x = -2 \), \( f(-2) = \frac{1}{2}(4)^{-2} = \frac{1}{2} \times \frac{1}{16} = \frac{1}{32} \).- For \( x = -1 \), \( f(-1) = \frac{1}{2}(4)^{-1} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \).- For \( x = 0 \), \( f(0) = \frac{1}{2}(4)^{0} = \frac{1}{2} \).- For \( x = 1 \), \( f(1) = \frac{1}{2}(4)^{1} = 2 \).- For \( x = 2 \), \( f(2) = \frac{1}{2}(4)^{2} = 8 \).

Step 3: Graph \( f(x) \) Using the Values

Plot the points \((-2, \frac{1}{32})\), \((-1, \frac{1}{8})\), \((0, \frac{1}{2})\), \((1, 2)\), and \((2, 8)\) on the graph. Connect these points with a smooth curve to represent the function \( f(x) = \frac{1}{2}(4)^{x} \). The curve should rise steeply as \( x \) increases.

Step 4: Finding the Reflection of \( f(x) \) About the \( x \)-Axis

The reflection of a function \( f(x) \) about the \( x \)-axis is given by \(-f(x)\). For the given function, the reflection is \(-f(x) = -\frac{1}{2}(4)^{x}\).

Step 5: Calculate Values for \(-f(x)\)

Using the same \( x \)-values from Step 2, calculate the corresponding \(-f(x)\) values.- For \( x = -2 \), \(-f(-2) = -\frac{1}{32} \).- For \( x = -1 \), \(-f(-1) = -\frac{1}{8} \).- For \( x = 0 \), \(-f(0) = -\frac{1}{2} \).- For \( x = 1 \), \(-f(1) = -2 \).- For \( x = 2 \), \(-f(2) = -8 \).

Step 6: Graph \(-f(x)\) Using the Values

Plot the points \((-2, -\frac{1}{32})\), \((-1, -\frac{1}{8})\), \((0, -\frac{1}{2})\), \((1, -2)\), and \((2, -8)\) on the same graph as \( f(x) \). Connect these points with a smooth curve which will mirror the curve of \( f(x) \) across the \( x \)-axis. Make sure both curves are clearly labeled.

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a variable appears in the exponent. In our example, the function is \( f(x) = \frac{1}{2}(4)^x \). Here, the base of the exponent is 4, which is greater than 1. This means that the function will exhibit exponential growth. As the value of \( x \) increases, the function grows rapidly at an increasing rate. Such functions are characterized by their J-shaped curve when graphed, starting off slowly and then accelerating upwards. This behavior is crucial to understand when sketching its graph, as it helps anticipate how the curve behaves over a range of \( x \) values. Exponential functions are encountered in many real-world scenarios like population growth and radioactive decay.

Reflection Across the X-Axis

Reflecting a function across the x-axis involves inverting its y-values. For the function \( f(x) = \frac{1}{2}(4)^x \), its reflection is expressed as \( -f(x) = -\frac{1}{2}(4)^x \). This transformation flips the original curve over the x-axis, so every point \( (x, y) \) becomes \( (x, -y) \). For example, if a point on the original graph is \( (2, 8) \), the reflected point will be \( (2, -8) \). This reflection helps in visualizing how the function behaves symmetrically relative to the axis, providing insight into the function’s properties and its potential real-world applications.

Plotting Points

Plotting points is a fundamental step in graphing functions. It involves selecting a series of x-values and calculating the corresponding y-values using the function's equation. For \( f(x) = \frac{1}{2}(4)^x \), chosen x-values were \(-2, -1, 0, 1,\) and \( 2 \). The corresponding y-values are calculated as follows:

• \( f(-2) = \frac{1}{32} \)
• \( f(-1) = \frac{1}{8} \)
• \( f(0) = \frac{1}{2} \)
• \( f(1) = 2 \)
• \( f(2) = 8 \)

These points are crucial as they form the skeleton for your graph. Begin plotting these on a Cartesian plane to visualize the function's progression. A smooth curve can then be drawn to connect these points, capturing the exponential relationship.

Curve Sketching

Curve sketching is the art of drawing a smooth curve through plotted points to represent a function's behavior visually. For the exponential function \( f(x) = \frac{1}{2}(4)^x \), after plotting the points, it's time to connect them smoothly, reflecting the rapid growth nature of exponential functions. The curve should start close to the x-axis when x is negative and rise rapidly as x becomes positive.

• Ensure the curve passes through each plotted point.
• The curve should be smooth, with no sharp angles.
• As for reflections, the curve of \(-f(x)\) needs to mirror this behavior under the x-axis.

This smooth and precise curve sketching provides a clear pictorial representation, enabling easier analysis and interpretation of the function's characteristics.