What Is The Equation Of The Translated Function

What is the Equation of a Translated Function? A Comprehensive Guide

Understanding how functions transform is crucial in mathematics, particularly in calculus and advanced algebra. This comprehensive guide explores the equation of a translated function, covering both horizontal and vertical shifts, and extending to more complex transformations involving stretches and reflections. We’ll delve into the underlying principles and provide numerous examples to solidify your understanding.

Understanding Function Transformations

A function, in essence, is a relationship between an input (typically 'x') and an output (typically 'y'). A translated function is simply a modified version of the original function, where its graph is shifted or moved on the coordinate plane without changing its fundamental shape. These shifts are achieved by altering the input ('x') and/or the output ('y') values.

Key Types of Transformations

The most common transformations are:

• Vertical Translation: Shifting the graph up or down along the y-axis.
• Horizontal Translation: Shifting the graph left or right along the x-axis.
• Vertical Stretch/Compression: Stretching or compressing the graph vertically, away from or towards the x-axis.
• Horizontal Stretch/Compression: Stretching or compressing the graph horizontally, away from or towards the y-axis.
• Reflection: Reflecting the graph across the x-axis or the y-axis.

Vertical Translation: Shifting Up and Down

A vertical translation shifts the entire graph of a function parallel to the y-axis. If we have a function f(x), a vertical translation by a constant 'k' units is represented as:

g(x) = f(x) + k

• k > 0: The graph shifts upwards by 'k' units.
• k < 0: The graph shifts downwards by 'k' units.

Example:

Let's consider the function f(x) = x². If we want to shift this parabola upward by 3 units, the translated function would be:

g(x) = f(x) + 3 = x² + 3

This new function, g(x), is identical in shape to f(x) but is positioned 3 units higher on the y-axis.

Horizontal Translation: Shifting Left and Right

A horizontal translation shifts the graph parallel to the x-axis. This transformation involves modifying the input value ('x') directly. The translated function is represented as:

g(x) = f(x - h)

• h > 0: The graph shifts to the right by 'h' units.
• h < 0: The graph shifts to the left by 'h' units (note the negative sign).

Example:

Consider the same function f(x) = x². To shift the parabola 2 units to the right, we use:

g(x) = f(x - 2) = (x - 2)²

This new parabola has the same shape but is positioned 2 units to the right of the original. Notice that to move right, we subtract from 'x', and to move left, we add to 'x'. This is often a source of confusion for beginners, so take careful note!

Combining Vertical and Horizontal Translations

We can combine both vertical and horizontal translations to create more complex shifts. The general form of a function translated both vertically and horizontally is:

g(x) = f(x - h) + k

Where:

• 'h' represents the horizontal shift.
• 'k' represents the vertical shift.

Example:

Let's translate f(x) = x² 4 units to the left and 1 unit down. The translated function would be:

g(x) = f(x + 4) - 1 = (x + 4)² - 1

Vertical and Horizontal Stretches and Compressions

Besides translations, functions can undergo stretches and compressions. These transformations alter the shape of the graph by scaling it along the x-axis or y-axis.

Vertical Stretch/Compression

A vertical stretch or compression is represented by multiplying the function by a constant 'a':

g(x) = a * f(x)

• |a| > 1: The graph is stretched vertically.
• 0 < |a| < 1: The graph is compressed vertically.
• a < 0: The graph is also reflected across the x-axis.

Horizontal Stretch/Compression

A horizontal stretch or compression involves multiplying the input 'x' by a constant 'b':

g(x) = f(bx)

• 0 < |b| < 1: The graph is stretched horizontally.
• |b| > 1: The graph is compressed horizontally.
• b < 0: The graph is reflected across the y-axis.

Example:

If f(x) = x³, then g(x) = 2f(x) = 2x³ represents a vertical stretch by a factor of 2, while g(x) = f(2x) = (2x)³ = 8x³ represents a horizontal compression by a factor of 1/2. Note the different effects on the overall graph.

Reflections: Mirroring the Graph

Reflections mirror the graph across either the x-axis or the y-axis.

Reflection across the x-axis

Reflecting across the x-axis involves multiplying the entire function by -1:

g(x) = -f(x)

Reflection across the y-axis

Reflecting across the y-axis involves replacing 'x' with '-x':

g(x) = f(-x)

Example:

If f(x) = √x, then g(x) = -√x is a reflection across the x-axis, and g(x) = √(-x) is a reflection across the y-axis. Note that the latter is only defined for negative x values.

Combining Multiple Transformations

In real-world applications, you often encounter functions that have undergone multiple transformations. The order in which these transformations are applied is crucial. Generally, horizontal transformations (shifts, stretches, reflections) are applied before vertical transformations.

Example:

Let's say we want to translate f(x) = x² 3 units to the right, stretch it vertically by a factor of 2, and then shift it 1 unit up. The order of operations would be:

1. Horizontal shift: f(x - 3) = (x - 3)²
2. Vertical stretch: 2f(x - 3) = 2(x - 3)²
3. Vertical shift: 2f(x - 3) + 1 = 2(x - 3)² + 1

The final translated function is g(x) = 2(x - 3)² + 1.

Working with Piecewise Functions

Piecewise functions, defined by different expressions across different intervals, also undergo transformations. Each piece of the function is transformed individually.

Example:

Consider a piecewise function:

f(x) = { x, x ≥ 0; -x, x < 0}

If we want to shift it 2 units to the right, the transformed function would be:

g(x) = { x - 2, x ≥ 2; -(x - 2), x < 2}

Each piece of the function is shifted according to the transformation rule.

Advanced Transformations and Applications

This guide has covered the fundamental aspects of function translations. More advanced topics include:

• Transformations of Trigonometric Functions: Understanding how phase shifts, amplitude changes, and period changes affect sine, cosine, and tangent graphs.
• Transformations in Polar Coordinates: Applying transformations to functions defined in polar coordinates.
• Applications in Calculus: Applying transformations to optimize functions and solve optimization problems.

Mastering function transformations is essential for a deeper understanding of mathematical concepts and their applications in various fields, including physics, engineering, and computer science. The ability to visualize and predict the effects of transformations is crucial for problem-solving and analysis. Remember, practice is key – the more examples you work through, the more comfortable you'll become with these powerful tools.