Problem 2 - Precalculus with Limits

Question

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 reflection of a function \(y = f(x)\) in the \(x\)-axis is represented by \(h(x) = -f(x)\), and the reflection in the \(y\)-axis is represented by \(h(x) = f(-x)\).

Step 1 - Reflect along the x-axis

A reflection of a function \(y = f(x)\) in the \(x\)-axis is achieved changing the sign of the y coordinate. This is done by taking the negative of the function. Hence, \(h(x) = -f(x)\)

Step 2 - Reflect along the y-axis

For the reflection of a function \(y = f(x)\) along the \(y\)-axis, the sign of the x coordinate is flipped. This is the equivalent of replacing every \(x\) in the function by \(-x\). Hence, \(h(x) = f(-x)\)

Key Concepts

x-axis reflection

When we talk about reflections in functions, one important transformation is the **x-axis reflection**. Imagine a mirror lying exactly on the x-axis of a graph, reflecting everything from the upside downwards to the other side of the axis. This transformation flips the graph of the function vertically.

For a given function \( y = f(x) \), reflecting it across the x-axis means changing the sign of every y-coordinate. What this means is every time a y-value appears in your function, you will multiply it by -1.

• Original: \( y = f(x) \)
• Reflection: \( h(x) = -f(x) \)

This simple multiplication by -1 can completely change the nature of the function.
If your original graph had a peak, now it has a trough, and vice versa. Understanding this can help you easily visualize how the graph of a function will look after the reflection.

y-axis reflection

While the x-axis reflection involves flipping the graph vertically, a **y-axis reflection** involves a horizontal transformation. Imagine now that the mirror is standing upright on the y-axis. This transformation changes the graph by reflecting it horizontally across the y-axis.

For a function \( y = f(x) \), conducting a reflection across the y-axis involves changing the sign of each instance of x in the function. This reflects all points horizontally to their counterparts on the opposite side of the y-axis.

• Original: \( y = f(x) \)
• Reflection: \( h(x) = f(-x) \)

This change of sign in x means that for each positive x-value, there is now a corresponding negative x-value that has the same y-value. All points on the graph are swapped from one side of the y-axis to the other, leaving the y-values unchanged, but effectively creating a mirror image along the y-axis.

transformations in functions

Beyond reflections, functions can undergo a variety of transformations. These **transformations in functions** include translations, stretches, and compressions as well. Each transformation changes the appearance of the graph in a unique way.

- **Translations**: Shifting the graph up, down, left, or right without altering its shape. - Up or Down: Add or subtract a constant from the function (\( y = f(x) + k \)). - Left or Right: Adjust the x-value inside the function (\( y = f(x + h) \)).- **Stretches and Compressions**: Alter the shape of the graph by scaling it. - Vertical Stretch/Compression: Multiply the whole function by a constant greater or less than 1 respectively. - Horizontal Stretch/Compression: Multiply the x-variable inside the function by a constant.
Understanding these transformations can help you manipulate graphs easily. Whether reflecting, translating, stretching, or compressing, each has a direct impact on the function's graph. Comprehensive mastery of these concepts will enable you to predict and draw transformed graphs quickly.