Problem 72
Question
Find a function \(f\) whose graph is the given curve \(\mathcal{C}\). \(\mathcal{C}\) is obtained by reflecting the graph of \(y=(x+1) /\) \(\left(x^{4}+1\right)\) about the origin.
Step-by-Step Solution
Verified Answer
The function is \(f(x) = \frac{x - 1}{x^4 + 1}\).
1Step 1: Understand Reflection About the Origin
When a graph is reflected about the origin, every point \((x, y)\) on the graph is transformed to \((-x, -y)\). That means if \( (x, y) \) lies on the original function, then \((-x, -y)\) will lie on the reflected function.
2Step 2: Determine the Reflected Function Points
Given the original function \( y = \frac{(x+1)}{(x^4+1)} \), reflect it about the origin by determining the reflection of each point on the curve. A point \((x, y)\) becomes \((-x, -y)\), which implies the equation \(-y = \frac{(-x+1)}{((-x)^4+1)} \).
3Step 3: Solve for the Reflected Function
Start with the transformed equation from Step 2: \(-y = \frac{(-x+1)}{(x^4+1)}\). To find the function \(f(x)\) representing the reflected curve, set \(y = \frac{x-1}{x^4+1}\). Therefore, the reflected function is \(f(x) = \frac{x - 1}{x^4 + 1}\).
Key Concepts
Graph TransformationOrigin ReflectionCoordinate Geometry
Graph Transformation
Graph transformation involves modifying the position, shape, or size of a graph, resulting in a new graph. This can include shifts, stretches, reflections, and more.
Reflections, like the one we explored in the exercise, are one type of graph transformation. A graph can be flipped over an axis or over the origin, producing a mirror image of the original graph. Understanding these transformations is crucial for analyzing how different functions relate to each other visually.
Other types of transformations include:
Reflections, like the one we explored in the exercise, are one type of graph transformation. A graph can be flipped over an axis or over the origin, producing a mirror image of the original graph. Understanding these transformations is crucial for analyzing how different functions relate to each other visually.
Other types of transformations include:
- Horizontal and vertical shifts: moving the graph up, down, left, or right.
- Stretching and compressing: altering the width or height of the graph by multiplying by a factor.
- Rotations: rotating the graph around a central point.
Origin Reflection
Reflecting a graph about the origin is a specific type of reflection. When reflecting about the origin, each point \((x, y)\) on the original graph changes to \((-x, -y)\) on the new graph. This transformation results in a graph that is symmetrically opposite across both the x-axis and y-axis.
Origin reflections essentially flip the graph both horizontally and vertically. In terms of algebra, it means replacing \(x\) with \(-x\) and \(y\) with \(-y\) in the equation of the function. The exercise shows this transformation explicitly by changing the function \(y = \frac{(x+1)}{(x^4+1)}\) to \(-y = \frac{(-x+1)}{((x)^4+1)}\).
This reflection technique is powerful for functions with symmetry through the origin, allowing us to identify functional relationships at a glance. Origin reflections are often used to analyze anti-symmetric functions and understand their properties.
Origin reflections essentially flip the graph both horizontally and vertically. In terms of algebra, it means replacing \(x\) with \(-x\) and \(y\) with \(-y\) in the equation of the function. The exercise shows this transformation explicitly by changing the function \(y = \frac{(x+1)}{(x^4+1)}\) to \(-y = \frac{(-x+1)}{((x)^4+1)}\).
This reflection technique is powerful for functions with symmetry through the origin, allowing us to identify functional relationships at a glance. Origin reflections are often used to analyze anti-symmetric functions and understand their properties.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics links algebra and geometry, allowing us to solve geometrical problems using algebraic methods.
A core benefit of understanding coordinate geometry is its ability to help visualize and solve problems involving shapes and graphs. For instance, in the exercise, reflecting a graph about the origin uses concepts from coordinate geometry to modify the function's visual representation by adjusting its algebraic equation.
Some fundamental concepts in coordinate geometry include:
A core benefit of understanding coordinate geometry is its ability to help visualize and solve problems involving shapes and graphs. For instance, in the exercise, reflecting a graph about the origin uses concepts from coordinate geometry to modify the function's visual representation by adjusting its algebraic equation.
Some fundamental concepts in coordinate geometry include:
- The Cartesian plane: a two-dimensional plane defined by an x-axis and a y-axis.
- Points: represented as a pair of numbers \((x, y)\) denoting their positions.
- Lines: equations defining straight paths on the plane.
- Curves and graphs: algebraic equations defining sets of points.
Other exercises in this chapter
Problem 71
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