Problem 73 - College Algebra with Modeling and Visualization

Question

For the given representation of a function \(f,\) graph the reflection across the \(x\)-axis and graph the reflection across the \(y\)-axis. Line graph determined by the table \(\begin{array}{rrrrr}x & -3 & -1 & 1 & 2 \\ f(x) & 2 & 3 & -1 & -2\end{array}\)

Step-by-Step Solution

Answer

Reflected across the x-axis: \((-3, -2), (-1, -3), (1, 1), (2, 2)\). Reflected across the y-axis: \((3, 2), (1, 3), (-1, -1), (-2, -2)\).

Step 1: Understand How Reflection Works

Reflection of a graph across the \(x\)-axis involves changing the sign of the \(y\)-coordinates of the function. That means each point \((x, f(x))\) in the original function will become \((x, -f(x))\) for the reflection. Similarly, reflection across the \(y\)-axis involves changing the sign of the \(x\)-coordinates; each point \((x, f(x))\) will become \((-x, f(x))\).

Step 2: Reflect Across the X-Axis

To get the reflection across the \(x\)-axis, take each \(f(x)\) value and multiply it by \(-1\). The table for \(f\) reflected across the \(x\)-axis is: \[\begin{array}{rrrrr} x & -3 & -1 & 1 & 2 \ -f(x) & -2 & -3 & 1 & 2\end{array}\]

Step 3: Reflect Across the Y-Axis

To create the reflection across the \(y\)-axis, take each \(x\) value and multiply it by \(-1\). The table for \(f\) reflected across the \(y\)-axis is: \[\begin{array}{rrrrr} -x & 3 & 1 & -1 & -2 \ f(x) & 2 & 3 & -1 & -2\end{array}\]

Step 4: Interpret the Results

The new coordinates representing the reflection across the \(x\)-axis are:\((-3, -2), (-1, -3), (1, 1), (2, 2)\). The coordinates for the reflection across the \(y\)-axis are: \((3, 2), (1, 3), (-1, -1), (-2, -2)\). These can be graphed by plotting the points on the appropriate axes.

Key Concepts

Coordinate Transformations

Coordinate transformations are a fundamental concept in geometry and algebra. They involve changing the position or orientation of a function or figure without altering its shape or size. These transformations can include translations, rotations, reflections, and scaling. Reflections, in particular, are transformations that "flip" a graph or figure over a specified axis, thus changing its position in a predictable way.

To perform a reflection, you need to choose an axis as your line of reflection. This axis acts like a mirror, where each point on the original function has a corresponding point on the opposite side of the axis. For coordinates, you change the sign of the coordinates accordingly:

• Reflection across the x-axis: Change the sign of the y-coordinates. The point \( (x, y) \) becomes \( (x, -y) \).
• Reflection across the y-axis: Change the sign of the x-coordinates. The point \( (x, y) \) becomes \( (-x, y) \).

These reflections help us understand symmetry and can be particularly useful in analyzing even and odd functions.

Graphing Functions

Graphing functions is a key skill in understanding how mathematical relationships work visually. When you graph a function, you're essentially creating a visual representation of all the possible outputs (\( f(x) \) values) for each input (\( x \) value). This allows you to see patterns, trends, and behaviors of functions at a glance.

To graph a function, follow these simple steps:

• Determine a set of x-values from the domain of the function.
• Calculate the corresponding y-values using the function rule \( f(x) \).
• Plot the points \( (x, f(x)) \) on a coordinate plane.
• Connect the points to reveal the shape of the function's graph.

Graphing provides insightful information and can be particularly helpful when analyzing the impact of transformations such as reflections. By plotting both the original and the transformed points, you can make direct visual comparisons and understand better how functions behave under different transformations.

Reflections Across Axes

Reflections across axes are specific coordinate transformations that greatly impact the appearance of a graph. Understanding how these reflections work allows you to predict and verify changes to graph orientations. When reflecting across the x-axis:

• For each point on the graph, you invert the sign of the y-coordinate.
• This shifts every point directly across the x-axis, creating a "mirror" image of the original graph.

For example, the point \((x, y)\) reflected across the x-axis becomes \((x, -y)\). This reflection affects both increasing and decreasing parts of the graph.

Similarly, a reflection across the y-axis involves taking the original graph points \((x, y)\) and changing the sign of the x-coordinate, making them \((-x, y)\). This flips the graph horizontally over the y-axis.

By understanding these approaches, you can comprehensively manipulate and interpret functions, enhancing your problem-solving skills in mathematics.