Problem 8 - College Algebra

Question

For the following exercises, graph the function and its reflection about the \(y\) -axis on the same axes, and give the \(y\) -intercept. $$ f(x)=3\left(\frac{1}{2}\right)^{x} $$

Step-by-Step Solution

Answer

The y-intercept is 3.

Step 1: Understand the Function

The function given is \( f(x) = 3 \left( \frac{1}{2} \right)^x \). This is an exponential decay function with base \( \frac{1}{2} \) and a vertical stretch factor of \( 3 \). We will graph this function first.

Step 2: Plot the Original Function

To graph \( f(x) = 3 \left( \frac{1}{2} \right)^x \), choose several values of \( x \) to plot points. For example, \( f(0) = 3 \), \( f(1) = 1.5 \), \( f(-1) = 6 \). Plot these points and join them to sketch the curve. It starts at 3 when \( x=0 \) and approaches 0 as \( x \rightarrow \infty \).

Step 3: Reflect the Function about the y-axis

The reflection of \( f(x) \) about the \( y \)-axis is given by \( f(-x) = 3 \left( \frac{1}{2} \right)^{-x} = 3 \cdot 2^x \). Plot this reflection by choosing points: \( f(-0) = 3 \), \( f(-1) = 6 \), \( f(1) = 1.5 \). Connect these points to graph the reflection which increases to the right.

Step 4: Identify the y-intercept

The \( y \)-intercept of a function is where \( x = 0 \). For both \( f(x) \) and its reflection \( f(-x) \), substitute \( x = 0 \) into both expressions: \( f(0) = 3 \) and \( f(-0) = 3 \). Therefore, the \( y \)-intercept of both functions is 3.

Key Concepts

Graphing Reflections

When graphing functions and their reflections, it is important to understand how the reflection affects the shape of the graph. Reflection over the y-axis means that for every point \((x, y)\) on the original function \(f(x)\), there is a corresponding point \((-x, y)\) on the reflection. This type of transformation essentially flips the graph horizontally.

Let's take the function \(f(x) = 3 \left( \frac{1}{2} \right)^x\). This is an exponential decay function, meaning it decreases as \(x\) increases. To find the reflection of this function over the y-axis, we replace \(x\) with \(-x\) in the function, transforming it to \(f(-x) = 3 \left( \frac{1}{2} \right)^{-x} = 3 \, \cdot \, 2^x\). This new function \(3 \, \cdot \, 2^x\) is an exponential growth function.

Plotting both these functions on the same set of axes helps visualize their reflection properties: the original decay function falls off as \(x\) becomes more positive, while its reflection grows as \(x\) becomes more positive.

Y-Intercept

Understanding the y-intercept of functions is crucial for graphing. The y-intercept is the point where the graph of a function crosses the y-axis. For any function \(f(x)\), this occurs when \(x = 0\).

In the function \(f(x) = 3 \left( \frac{1}{2} \right)^x\), the y-intercept can be found by substituting \(x = 0\), giving \(f(0) = 3 \left( \frac{1}{2} \right)^0 = 3\). Thus, the point \((0, 3)\) is the y-intercept of this function. Similarly, for the reflection \(f(-x) = 3 \cdot 2^x\), the y-intercept is also at \((0, 3)\).

Both the original function and its reflection share the same y-intercept, which indicates they both pass through the same point on the y-axis. This is a useful feature to know when trying to quickly sketch this pair of related functions.

Visualizing Transformations

Visualizing transformations involves understanding how different algebraic changes affect the graphical representation of a function. Key transformations include reflections, translations, dilations, and rotations. In the context of exponential functions, reflections and dilations are particularly prevalent.

For the exponential decay function \(f(x) = 3 \left( \frac{1}{2} \right)^x\), its transformation to \(f(-x) = 3 \cdot 2^x\) demonstrates a reflection over the y-axis, turning a decay function into a growth one. The coefficient \(3\) acts as a vertical stretch, pulling the graph away from the x-axis, making it taller.

Graphing these transformations provides clear visual feedback on how equations translate into shapes and patterns on a graph. By choosing several points and plotting both \(f(x)\) and \(f(-x)\), students can observe how each transformation manifests geometrically, reinforcing their understanding of these concepts in a vivid and practical manner.