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.