Problem 2 - Precalculus

Question

Fill in the blanks. A reflection in the \(x\)-axis of \(y=f(x)\) is represented by \(h(x)\)=_____ while a reflection in the \(y\)-axis of \(y=f(x)\) is represented by \(h(x)\)=_____.

Step-by-Step Solution

Answer

The answer to the exercise is: A reflection in the \(x\)-axis of \(y=f(x)\) is represented by \(h(x)=-f(x)\) while a reflection in the \(y\)-axis of \(y=f(x)\) is represented by \(h(x)=f(-x)\).

Step 1: Reflect over the x-axis

A reflection of a function over the x-axis implies that the signs of the y-values of the function are reversed. Hence if \(y=f(x)\) is the given function, a reflection in the x-axis would be represented by the function \(h(x)=-f(x)\). Hence \(h(x)=-f(x)\) is the solution for the first blank in the exercise.

Step 2: Reflect over the y-axis

A reflection over the y-axis implies that the signs of the x-values of the function are reversed. Hence if \(y=f(x)\) is the given function, a reflection in the y-axis would be represented by the function \(h(x)=f(-x)\). Hence \(h(x)=f(-x)\) is the solution for the second blank in the exercise.

Key Concepts

x-axis reflection

Reflecting a function over the x-axis means flipping the graph of the function upside down. This process changes the sign of all the y-values while keeping the x-values the same. If you have a function denoted as \( y = f(x) \), its reflection across the x-axis is expressed as \( h(x) = -f(x) \).

This transformation takes each point on the original function and moves it to an equivalent point on the opposite side of the x-axis. For example, if a point on the original function is (2, 3), the reflected point will be (2, -3).

The result of an x-axis reflection results in an image that is a mirror version of the original graph, flipped top to bottom. Here are some key takeaways:

• Only the y-coordinates are affected; they are multiplied by -1.
• The graph's shape remains the same, just flipped over the x-axis.
• Reflections over the x-axis are used in solving equations and observing symmetry in functions.

y-axis reflection

In a y-axis reflection, the function's graph is flipped left to right. This involves changing the sign of the x-values, while the y-values remain unchanged. Mathematically, the reflection of a function \( y = f(x) \) over the y-axis is given by \( h(x) = f(-x) \).

This means every x-coordinate in the function is reversed, so the point (a, b) on the original graph becomes (-a, b) on the reflected graph. This creates a mirror image of the function along the y-axis.

This type of reflection can help in understanding properties of symmetry and how a function behaves with respect to the vertical axis.

• Only x-coordinates change sign; y-coordinates remain the same.
• Useful for studying even functions that are symmetrical about the y-axis.
• Affects the horizontal orientation of the graph.

function transformation

Function transformations encompass a wide range of operations that alter a function's graph, including reflections, translations, stretches, and compressions.

Each transformation type manipulates the graph in a unique way. For instance, reflections flip the graph over an axis, translations slide the graph to different locations, while stretches and compressions change the graph's size.

Understanding these transformations is crucial for gaining insight into the behavior of different functions.

• Reflections: As discussed, include flipping over the x-axis (\( h(x) = -f(x) \)) and y-axis (\( h(x) = f(-x) \)).
• Translations: Move the graph up, down, left, or right by adjusting the function by adding or subtracting constants.
• Stretches/Compressions: Change the graph's slope by multiplying the function by constants greater than or less than one.

These transformations allow mathematicians and students to manipulate functions for better understanding or solving complex problems.