Problem 8 - Precalculus

Question

Suppose (2,-3) is on the graph of \(y=f(x) .\) In Exercises \(1-18,\) use Theorem 1.7 to find a point on the graph of the given transformed function. $$ y=f(-x) $$

Step-by-Step Solution

Answer

The point \((-2, -3)\) is on the graph of \(y = f(-x)\).

Step 1: Identifying the Original Point

Start by noting the given point \((2, -3)\), which lies on the graph of \(y = f(x)\). This means that when \(x = 2\), the value of \(y = f(x) = -3\).

Step 2: Analyzing Transformation

The transformation we are considering is \(y = f(-x)\). This transformation reflects the graph of \(y = f(x)\) across the y-axis. To find a point on this transformed graph, replace \(x\) with \(-x\) in the input.

Step 3: Finding the Transformed Point

Given the point \((2, -3)\) on the original graph, for the graph of \(y = f(-x)\), replace \(x = 2\) with \(-2\). This results in the new point \((-2, -3)\) on the graph of \(y = f(-x)\).

Key Concepts

Graph Reflection

Graph reflection is a fundamental concept in mathematics that deals with mirroring the graph of a function across an axis. Think of it as flipping the image of the graph over a line, similar to how a reflection in a mirror works. In the context of functions, graph reflections help us visualize how graphs change and behave with various transformations.

• Reflection can be across the x-axis, y-axis, or any other line.
• The reflection changes the position of each point on the graph, creating a mirror image.
• It doesn't alter the shape or size of the graph, only its orientation.

Reflecting a graph across different lines or axes can affect the way a function is understood and interpreted, especially in complex functions. Comprehending these transformations aids in deeper mathematical insight.

Y-axis Reflection

Y-axis reflection specifically focuses on flipping the graph of a function over the y-axis. When you perform a y-axis reflection, every point of the graph on the right side of the y-axis is mirrorred to the left, and vice versa.To achieve this mathematically, you simply replace every instance of the variable within the function with its negative:

• This means if your original function is expressed as \(y = f(x)\), the reflected function will be \(y = f(-x)\).
• The x-coordinates change their signs, but the y-coordinates remain the same.
• Example: If a point on the original graph is \( (2, -3) \), its reflection will be at \( (-2, -3) \).

Understanding this reflection offers a window into how symmetry works in functions, which is crucial for analyzing and solving equations more effectively.

Input Substitution

Input substitution is an essential process used during the transformation of functions, particularly when dealing with reflections. This method involves switching the input value within a function to a new value according to the transformation rule.

• For y-axis reflection, we carry out input substitution by changing \(x\) to \(-x\) in the function's equation.
• This substitution affects all points on the function and hence alters the graph entirely.
• For example, for a point \( (2, -3) \) on a function \(y = f(x)\), the input substitution in a reflected function \(y = f(-x)\) results in the point \( (-2, -3) \).

Understanding input substitution allows students to grasp how simple alterations can have a significant impact on the behavior and appearance of a function's graph.

Function Graph Points

Function graph points are essential for plotting and understanding mathematical functions. Each point, defined by an \((x, y)\) coordinate, represents a specific solution to the function's equation. When a transformation is applied, these coordinates shift, providing insight into the nature of the transformation.

• The graph of a function is a collection of these points plotted in a coordinate plane.
• Transformations change these points' locations depending on whether they involve reflections, translations, or other operations.
• In exercises involving transformations, like the y-axis reflection, identifying these points helps in visualizing how the graph morphs.

By focusing on individual graph points during transformations, students can better understand the holistic changes occurring in the graph, making these concepts clearer and more accessible.