AOS1 Topic 10: Transformation of Functions
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 point a units to the right and b units up.
\[(x, y) → (x + a, y + b)\]
Reflection: Flipping a shape over a line (the axis of reflection). For example, reflecting a shape over the y-axis.
\[(x, y) → (−x, y)\]
Scaling (Dilation): Resizing a shape by a certain factor. For example, enlarging a circle by a factor of k.
\[(x, y) → (kx, ky)\]
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.
\[f(x) → a f(x)\]
a > 1 stretches,
0 < a < 1 compresses
Horizontal Stretch/Compression: Stretching or compressing the graph horizontally.
\[f(x) → f(bx)\]
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
In the Cartesian plane \( \mathbb{R}^2 = \{ (x, y) : x, y \in \mathbb{R} \} \). a translation is a transformation that shifts every point the same distance in the same direction.
Notation and Rules
Translation by $(h,k)$
Moving all points $h$ units in the $x$-direction and $k$ units in the $y$-direction is described by
\[(x,y) \rightarrow (x+h,y+k)\]
Equivalently, we write $x' = x+h$ and $y' = y+k$
Example
Translating by 2 units right and 4 units up:
This can be described by the rule \((x, y) \to (x + 2, y + 4)\). This reads as ‘\((x, y)\) maps to \((x + 2, y + 4)\)’.
Effects on Graphs
If a curve is described by $y=f(x)$, then applying the translation $(x,y) \to (x+h, y+k)$ is equivalent to replacing $x$ with $(x - h)$ and $y$ with $(y−k)$ in the original equation.
\[y-k = f(x-h)\]
Dilations
A dilation stretches or shrinks a graph away from or toward a fixed axis by a constant factor.
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 \((x, y) \rightarrow (x, 2y)\).
Rule : $(x,y) → (x, b y)$.
Effect on Equations : For $y=f(x)$, replacing $y$ with $\frac{y}{b}$ by yields:
\[ \frac{y}{b} = f(x) \Longrightarrow y = b f(x)\].
Hence the point with coordinates (1, 1) → (1, 2).
Example :
A dilation of factor $2$ from the $x$-axis sends $(x, y)$ to $(x, 2y)$. The curve $y = \sqrt{x}$ becomes $y = 2\sqrt{x}$.
Dilation from the y-axis
A dilation of factor 2 from the y-axis is defined by the rule \((x, y) \rightarrow (2x, y)\).
Rule : $(x,y) → (ax, y)$.
Effect on Equations : For $y=f(x)$, replacing $x$ with $\frac{x}{a}$ by yields:
\[y = f( \frac{x}{a})\].
Hence the point with coordinates (1, 1) → (2, 1).
Example :
A dilation of factor $2$ from the $y$-axis sends $(x, y)$ to $(2x, y)$. The curve $y = \sqrt{x}$ becomes $y = \sqrt{\frac{x}{2}}$.
Reflections
Reflections studied here are across the $x$-axis or $y$-axis. (Reflections across the line $y=x$ are typically used to find the inverse relation and are addressed separately.)
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 \((1, 1) \rightarrow (1, -1)\).
Rule : $(x,y) → (x, −y)$.
Effect on Graphs : For $y=f(x)$, replace $y$ with $−y$, giving $y=− f(x)$. Hence every point $(x, f(x))$ is reflected to $(x, −f(x))$.
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 \((1, 1) \rightarrow (-1, 1)\).
Rule : $(x,y) → (-x, y)$.
Effect on Graphs : For $y=f(x)$, replace $x$ with $−x$, giving $y= f(-x)$. Hence every point $(x, f(x))$ is reflected to $(x, f(-x))$.