Function Transformations: \(y = Af(n(x+b)) + c\)

Overview

This section covers transformations of functions of the form \(y = f(x)\) to \(y = Af(n(x+b)) + c\), where \(A, n, b,\) and \(c\) are real numbers, and \(A, n eq 0\). We’ll also explore the inverse transformation.

Key Concepts

• Transformations: Alterations to a function’s graph, including dilations (stretches/compressions), reflections, and translations.
• Dilation: Stretching or compressing a graph. A dilation from the x-axis is a vertical stretch/compression, and a dilation from the y-axis is a horizontal stretch/compression.
• Reflection: Flipping a graph over a line (x-axis or y-axis).
• Translation: Shifting a graph horizontally or vertically.

The General Transformation: \(y = Af(n(x+b)) + c\)

This equation combines several transformations. Let’s break down each parameter:

• A: Vertical dilation (stretch/compression) and/or reflection in the x-axis.

  • If \(|A| > 1\): Vertical stretch by a factor of \(|A|\).
  • If \(0 < |A| < 1\): Vertical compression by a factor of \(|A|\).
  • If \(A < 0\): Reflection in the x-axis.

• n: Horizontal dilation (stretch/compression) and/or reflection in the y-axis.

  • If \(|n| > 1\): Horizontal compression by a factor of \(\frac{1}{|n|}\).
  • If \(0 < |n| < 1\): Horizontal stretch by a factor of \(\frac{1}{|n|}\).
  • If \(n < 0\): Reflection in the y-axis.

• b: Horizontal translation.

  • If \(b > 0\): Translation to the left by \(b\) units.
  • If \(b < 0\): Translation to the right by \(|b|\) units.

• c: Vertical translation.

  • If \(c > 0\): Translation upwards by \(c\) units.
  • If \(c < 0\): Translation downwards by \(|c|\) units.

Order of Transformations

It is crucial to apply transformations in the correct order. A recommended order is:

1. Dilations and Reflections: Apply vertical (A) and horizontal (n) dilations/reflections first.
2. Translations: Apply horizontal (b) and vertical (c) translations last.

Mapping Notation

We can represent the transformation using mapping notation:

\((x, y) \rightarrow (\frac{x}{n} - b, Ay + c)\)

This notation shows how a point \((x, y)\) on the original graph \(y = f(x)\) is mapped to a new point \((\frac{x}{n} - b, Ay + c)\) on the transformed graph \(y = Af(n(x+b)) + c\).

Inverse Transformations

To find the inverse transformation, we need to reverse the process. If the original transformation is:

\(y = Af(n(x+b)) + c\)

Then the inverse transformation can be found as follows:

1. Subtract \(c\) from both sides: \(y - c = Af(n(x+b))\)
2. Divide by \(A\): \(\frac{y-c}{A} = f(n(x+b))\)
3. Apply the inverse of \(f\), denoted as \(f^{-1}\): \(f^{-1}(\frac{y-c}{A}) = n(x+b)\)
4. Divide by \(n\): \(\frac{1}{n}f^{-1}(\frac{y-c}{A}) = x + b\)
5. Subtract \(b\): \(\frac{1}{n}f^{-1}(\frac{y-c}{A}) - b = x\)

Finally, swap \(x\) and \(y\) to express the inverse function in terms of x:

\(y = \frac{1}{n}f^{-1}(\frac{x-c}{A}) - b\)

Or, using mapping notation, the inverse transformation is:

\((x, y) \rightarrow (\frac{x - c}{A}, ny + b)\)

Examples

Example 1: \(y = 2(x-1)^2 + 3\)

• Original function: \(y = x^2\)
• \(A = 2\): Vertical stretch by a factor of 2.
• \(n = 1\): No horizontal dilation or reflection.
• \(b = -1\): Horizontal translation 1 unit to the right.
• \(c = 3\): Vertical translation 3 units up.

Example 2: \(y = -\sqrt{-x} + 1\)

• Original function: \(y = \sqrt{x}\)
• \(A = -1\): Reflection in the x-axis.
• \(n = -1\): Reflection in the y-axis.
• \(b = 0\): No horizontal translation.
• \(c = 1\): Vertical translation 1 unit up.

Common Functions and Transformations

Key Takeaways

• Understand the effect of each parameter (\(A, n, b, c\)) on the graph of \(y = f(x)\).
• Apply transformations in the correct order (dilations/reflections before translations).
• Use mapping notation to track the transformation of points.
• Be able to find and apply the inverse transformation.
• Practice with various functions and transformations to solidify your understanding.