AOS1 Topic 10: Transformation of Functions

In mathematics, a transformation refers to an operation that moves or changes a geometric figure or algebraic object in some way while preserving certain properties. Transformations can be applied to functions, shapes, and other mathematical objects, and they are fundamental in various areas of mathematics, including geometry, algebra, and calculus.

Types of Transformations

1. Geometric Transformations

Geometric transformations involve moving or changing a shape or figure in the plane or space. Common types include:

• Translation: Shifting a shape by a certain distance in a given direction. For example, moving a triangle 3 units to the right.
  (x, y) → (x + a, y + b)
• Rotation: Rotating a shape around a fixed point by a certain angle. For example, rotating a square 90 degrees around its center.
  (x, y) → (x cos θ − y sin θ, x sin θ + y cos θ)
• Reflection: Flipping a shape over a line (the axis of reflection). For example, reflecting a shape over the y-axis.
  (x, y) → (−x, y) (reflection over the y-axis)
• Scaling (Dilation): Resizing a shape by a certain factor. For example, enlarging a circle by a factor of 2.
  (x, y) → (kx, ky) (scaling by factor k)
• Shear: Slanting a shape in a specific direction. For example, shearing a rectangle so that it becomes a parallelogram.
  (x, y) → (x + ky, y) (horizontal shear)

2. Algebraic Transformations

In the context of functions, transformations refer to operations that alter the graph of a function. Common transformations include:

• Vertical Shift: Moving the graph up or down.
  f(x) → f(x) + c
• Horizontal Shift: Moving the graph left or right.
  f(x) → f(x - c)
• Vertical Stretch/Compression: Stretching or compressing the graph vertically.
  f(x) → af(x) (where a > 1 stretches, 0 < a < 1 compresses)
• Horizontal Stretch/Compression: Stretching or compressing the graph horizontally.
  f(x) → f(bx) (where b > 1 compresses, 0 < b < 1 stretches)
• Reflection: Flipping the graph over an axis.
  f(x) → -f(x) (reflection over the x-axis)
  f(x) → f(-x) (reflection over the y-axis)

Now we will discuss in detail:

Translations

The Cartesian plane is represented by the set \( \mathbb{R}^2 \) of all ordered pairs of real numbers. That is, \( \mathbb{R}^2 = \{ (x, y) : x, y \in \mathbb{R} \} \). The transformations considered in this book associate each ordered pair of \( \mathbb{R}^2 \) with a unique ordered pair. We can refer to them as examples of transformations of the plane.

For example, the translation 3 units in the positive direction of the x-axis (to the right) associates with each ordered pair (x, y) a new ordered pair (x + 3, y). This translation is a transformation of the plane. Each point in the plane is mapped to a unique second point. Furthermore, every point in the plane is an image of another point under this translation.

Notation

Consider the translation 2 units in the positive direction of the x-axis (to the right) and 4 units in the positive direction of the y-axis (up). This can be described by the rule \((x, y) \to (x + 2, y + 4)\). This reads as ‘\((x, y)\) maps to \((x + 2, y + 4)\)’.

For example, \((3, 2) \to (3 + 2, 2 + 4)\).

In applying this translation, it is useful to think of every point \((x, y)\) in the plane as being mapped to a new point \((x', y')\). This point \((x, y)\) is the only point which maps to \((x', y')\). The following can be written for this translation:

\(x' = x + 2\) and \(y' = y + 4\)

A translation of \(h\) units in the positive direction of the x-axis and \(k\) units in the positive direction of the y-axis is described by the rule:

\((x, y) \to (x + h, y + k)\)

\(x' = x + h\) and \(y' = y + k\)

where \(h\) and \(k\) are positive numbers.

A translation of \(h\) units in the negative direction of the x-axis and \(k\) units in the negative direction of the y-axis is described by the rule:

\((x, y) \to (x - h, y - k)\)

\(x' = x - h\) and \(y' = y - k\)

where \(h\) and \(k\) are positive numbers.

Notes:

• Under a translation, if \((a_0, b_0) = (c_0, d_0)\), then \((a, b) = (c, d)\).
• For a translation \((x, y) \to (x + h, y + k)\), for each point \((a, b) \in \mathbb{R}^2\) there is a point \((p, q)\) such that \((p, q) \to (a, b)\). (It is clear that \((p - h, q - k) \to (p, q)\) under this translation.)

Applying Translations to Sketch Graphs

A translation moves every point on the graph the same distance in the same direction.

Translations Parallel to an Axis

We start by looking at the images of the graph of \(y = x^2\) shown on the right under translations that are parallel to an axis.

General Translations of a Curve

Every translation of the plane can be described by giving two components:

• A translation parallel to the x-axis
• A translation parallel to the y-axis

Consider a translation of 2 units in the positive direction of the x-axis and 4 units in the positive direction of the y-axis applied to the graph of \(y = x^2\).

Translate the set of points defined by the function

\(\{ (x, y) : y = x^2 \}\)

by the translation defined by the rule

\((x, y) \rightarrow (x + 2, y + 4)\)
\(x' = x + 2\) and \(y' = y + 4\)

For each point \((x, y)\) there is a unique point \((x', y')\) and vice versa.

We have \(x = x' - 2\) and \(y = y' - 4\).

Translation of Curves

This means the points on the curve with equation \(y = x^2\) are mapped to the curve with equation \(y' - 4 = (x' - 2)^2\).

Example Points:

• (0, 0)
• (-1, 1)
• (1, 1)
• (2, 4)
• (1, 5)
• (3, 5)

Hence \(\{ (x, y) : y = x^2 \}\) maps to \(\{ (x', y') : y' - 4 = (x' - 2)^2 \}\).

For the graph of \(y = f(x)\), the following two processes yield the same result:

• Applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the graph of \(y = f(x)\).
• Replacing \(x\) with \(x - h\) and \(y\) with \(y - k\) in the equation to obtain \(y - k = f(x - h)\) and graphing the result.

Proof

A point \((a, b)\) is on the graph of \(y = f(x)\)

\(\Leftrightarrow f(a) = b\)

\(\Leftrightarrow f(a + h - h) = b\)

\(\Leftrightarrow f(a + h - h) = b + k - k\)

\(\Leftrightarrow (a + h, b + k)\) is a point on the graph of \(y - k = f(x - h)\)

Note: The double arrows indicate that the steps are reversible.

Dilations

We start with the example of a circle, as it is easy to visualize the effect of a dilation from an axis.

A dilation of a graph can be thought of as the graph ‘stretching away from’ or ‘shrinking towards’ an axis.

Dilation from the x-axis

We can determine the equation of the image of a curve under a dilation by following the same approach used for translations.

A dilation of factor 2 from the x-axis is defined by the rule \((x, y) \rightarrow (x, 2y)\).

Hence the point with coordinates (1, 1) → (1, 2).

Consider the curve with equation \( y = \sqrt{x} \) and the dilation of factor 2 from the x-axis.

• Let \((x', y')\) be the image of the point with coordinates \((x, y)\) on the curve.
• Hence \( x' = x \) and \( y' = 2y \), and thus \( x = x' \) and \( y = \frac{y'}{2} \).
• Substituting for \( x \) and \( y \), we see that the curve with equation \( y = \sqrt{x} \) maps to the curve with equation \( \frac{y'}{2} = \sqrt{x'} \), i.e., the curve with equation \( y = 2\sqrt{x} \).

For \( b \) a positive constant, a dilation of factor \( b \) from the x-axis is described by the rule:

\((x, y) \rightarrow (x, by)\)

Where \( x' = x \) and \( y' = by \).

For the graph of \( y = f(x) \), the following two processes yield the same result:

• Applying the dilation from the x-axis \((x, y) \rightarrow (x, by)\) to the graph of \( y = f(x) \).
• Replacing \( y \) with \( \frac{y}{b} \) in the equation to obtain \( y = b f(x) \) and graphing the result.

Dilation from the y-axis

A dilation of factor 2 from the y-axis is defined by the rule \((x, y) \rightarrow (2x, y)\). Hence the point with coordinates (1, 1) → (2, 1).

Again, consider the curve with equation \( y = \sqrt{x} \).

• Let \((x', y')\) be the image of the point with coordinates \((x, y)\) on the curve.
• Hence \( x' = 2x \) and \( y' = y \), and thus \( x = \frac{x'}{2} \) and \( y = y' \).
• The curve with equation \( y = \sqrt{x} \) maps to the curve with equation \( y' = \sqrt{\frac{x'}{2}} \).

For \( a \) a positive constant, a dilation of factor \( a \) from the y-axis is described by the rule:

\((x, y) \rightarrow (ax, y)\)

Where \( x' = ax \) and \( y' = y \).

For the graph of \( y = f(x) \), the following two processes yield the same result:

• Applying the dilation from the y-axis \((x, y) \rightarrow (ax, y)\) to the graph of \( y = f(x) \).
• Replacing \( x \) with \( \frac{x}{a} \) in the equation to obtain \( y = f \left( \frac{x}{a} \right) \) and graphing the result.

Dilation from the y-axis

A dilation of factor 2 from the y-axis is defined by the rule \((x, y) \rightarrow (2x, y)\). Hence the point with coordinates (1, 1) → (2, 1).

Again, consider the curve with equation \( y = \sqrt{x} \).

• Let \((x', y')\) be the image of the point with coordinates \((x, y)\) on the curve.
• Hence \( x' = 2x \) and \( y' = y \), and thus \( x = \frac{x'}{2} \) and \( y = y' \).
• The curve with equation \( y = \sqrt{x} \) maps to the curve with equation \( y' = \sqrt{\frac{x'}{2}} \).

For \( a \) a positive constant, a dilation of factor \( a \) from the y-axis is described by the rule:

\((x, y) \rightarrow (ax, y)\)

Where \( x' = ax \) and \( y' = y \).

For the graph of \( y = f(x) \), the following two processes yield the same result:

• Applying the dilation from the y-axis \((x, y) \rightarrow (ax, y)\) to the graph of \( y = f(x) \).
• Replacing \( x \) with \( \frac{x}{a} \) in the equation to obtain \( y = f \left( \frac{x}{a} \right) \) and graphing the result.

Reflections

The special case where the graph of a function is reflected in the line \( y = x \) to produce the graph of the inverse relation is discussed separately in Section 1F.

In this chapter we study reflections in the x- or y-axis only.

First consider reflecting the graph of the function shown here in each axis, and observe the effect on a general point \((x, y)\) on the graph.

Reflection in the x-axis

A reflection in the x-axis can be defined by the rule \((x, y) \rightarrow (x, -y)\). Hence the point with coordinates \((1, 1) \rightarrow (1, -1)\).

• Let \((x', y')\) be the image of the point \((x, y)\).
• Hence \(x' = x\) and \(y' = -y\), which gives \(x = x'\) and \(y = -y'\).
• The curve with equation \(y = \sqrt{x}\) maps to the curve with equation \(-y' = \sqrt{x'}\), i.e. the curve with equation \(y = -\sqrt{x}\).

A reflection in the x-axis is described by the rule \((x, y) \rightarrow (x, -y)\), where \(x' = x\) and \(y' = -y\).

For the graph of \(y = f(x)\), the following two processes yield the same result:

• Applying the reflection in the x-axis \((x, y) \rightarrow (x, -y)\) to the graph of \(y = f(x)\).
• Replacing \(y\) with \(-y\) in the equation to obtain \(y = -f(x)\) and graphing the result.

Reflection in the y-axis

A reflection in the y-axis can be defined by the rule \((x, y) \rightarrow (-x, y)\). Hence the point with coordinates \((1, 1) \rightarrow (-1, 1)\).

• Let \((x', y')\) be the image of the point \((x, y)\).
• Hence \(x' = -x\) and \(y' = y\), which gives \(x = -x'\) and \(y = y'\).
• The curve with equation \(y = \sqrt{x}\) maps to the curve with equation \(y' = \sqrt{-x'}\), i.e. the curve with equation \(y = \sqrt{-x}\).

A reflection in the y-axis is described by the rule \((x, y) \rightarrow (-x, y)\), where \(x' = -x\) and \(y' = y\).

For the graph of \(y = f(x)\), the following two processes yield the same result:

• Applying the reflection in the y-axis \((x, y) \rightarrow (-x, y)\) to the graph of \(y = f(x)\).
• Replacing \(x\) with \(-x\) in the equation to obtain \(y = f(-x)\) and graphing the result.