Problem 71 - Calculus Single Variable

Question

Find a function \(f\) whose graph is the given curve \(\mathcal{C}\). \(\mathcal{C}\) is obtained by reflecting the graph of \(y=\left(x^{3}+1\right)\) / \(\left(x^{2}+1\right)\) about the \(y\) -axis.

Step-by-Step Solution

Answer

The function is \( f(x) = \frac{-x^3 + 1}{x^2 + 1} \) after reflecting about the y-axis.

Step 1: Identify the Transformation

The exercise asks us to find a function that reflects the given function about the y-axis. When a graph is reflected about the y-axis, the x-coordinates of all points on the graph are multiplied by -1.

Step 2: Consider the Original Function

The original function given is \( y = \frac{x^3 + 1}{x^2 + 1} \). To reflect this graph about the y-axis, we need to substitute \( x \) with \( -x \) in the function.

Step 3: Apply the Reflection Transformation

By replacing \( x \) with \( -x \) in the original function, we obtain the new function: \( y = \frac{(-x)^3 + 1}{(-x)^2 + 1} \). Simplifying, we get \( y = \frac{-x^3 + 1}{x^2 + 1} \).

Step 4: Simplify the Result

The simplified function that represents the reflected graph is \( f(x) = \frac{-x^3 + 1}{x^2 + 1} \). This function is the reflection of the original function about the y-axis.

Key Concepts

Reflection

When we discuss function transformations, reflection is a fascinating concept. Reflection refers to flipping a graph across a specific axis. In this scenario, we're focusing on reflecting a function about the y-axis. To carry out this transformation, you adjust the x-coordinates of each point by multiplying them by -1.
This transformation inverts the graph horizontally, as if it's being viewed in a mirror placed along the y-axis. For instance, if a point on the original graph has coordinates (a, b), its reflected point will be (-a, b). This results in the entire curve flipping over the y-axis, changing its orientation.
In function notation, reflecting a function like \( y = f(x) \) about the y-axis entails replacing \( x \) with \( -x \). This gives the new, reflected function \( y = f(-x) \). The process of reflection is crucial in understanding how functions behave under transformations.

Y-Axis

The y-axis holds a central role in coordinate geometry as it serves as the vertical line that runs through the point where x equals zero. It divides the plane into two equal halves. Reflections, such as the one discussed in this exercise, often use the y-axis as a mirror line.
Understanding the y-axis' properties is crucial for graphing and reflecting functions. The y-axis is fundamental to plotting points and analyzing functions because every point on it fulfills the condition \( x = 0 \). When reflecting a function across this line, the y-coordinates remain unchanged, but the x-coordinates flip sign, as described earlier.
It is essential to grasp this concept as many transformations, especially in precalculus and calculus, rely on reflecting functions either across the y-axis or another designated line. Appreciating the significance of the y-axis assists in predicting how reflections and other transformations will affect a function's graph.

Graph Symmetry

A graph is said to be symmetric if it mirrors itself in some respect, indicating a repetitive or balanced pattern. Graph symmetry can take on various forms: about the y-axis, x-axis, or the origin. In particular, y-axis symmetry implies that if the point (x, y) is on the graph, then the point (-x, y) is also on the graph.
Symmetrical graphs about the y-axis don't change appearance when flipped horizontally. This kind of symmetry equals having a mirrored image on both sides of the y-axis. A common example would be the graph of a parabola opening upwards or downwards, such as \( y = x^2 \).

• Y-axis symmetry: Reflecting across the y-axis results in the same graph.
• X-axis symmetry: Reflecting across the x-axis results in the same graph.
• Origin symmetry: Rotating 180 degrees around the origin returns the same graph.

Understanding graph symmetry is vital for solving many geometric and algebraic problems. It simplifies graph analysis and is instrumental in identifying specific function properties.