Study Guide: Transformations of Functions in Precalculus

Transformations of Functions

Introduction to Transformations

Transformations occur when a function is manipulated and changes position and/or shape on the coordinate plane. Understanding transformations is essential for graphing and analyzing functions in Precalculus.

• Types of transformations: Reflection, Shift, Stretch/Shrink

• Transformations can be combined to produce complex changes in a function's graph.

Reflections

Reflection Over Axes

A reflection is a transformation that 'flips' a function over a specified axis.

• Reflection over the x-axis: The function $f(x)$ becomes $-f(x)$. All y-values change sign.

• Reflection over the y-axis: The function $f(x)$ becomes $f(-x)$. All x-values change sign.

Type	Transformation	Effect
Reflection over x-axis	$y = -f(x)$	Flips graph over x-axis
Reflection over y-axis	$y = f(-x)$	Flips graph over y-axis

Example: If $f(x) = x + 2$, then $g(x) = -f(x) = -(x + 2) = -x - 2$ is the reflection over the x-axis.

Shifts (Translations)

Vertical and Horizontal Shifts

A shift occurs when a function is moved vertically and/or horizontally from its original position.

• Vertical shift: $f(x) + k$ shifts the graph up ($k > 0$) or down ($k < 0$).

• Horizontal shift: $f(x - h)$ shifts the graph right ($h > 0$) or left ($h < 0$).

Type	Transformation	Effect
Vertical Shift	$y = f(x) + k$	Up if $k > 0$, Down if $k < 0$
Horizontal Shift	$y = f(x - h)$	Right if $h > 0$, Left if $h < 0$

Example: $f(x) = x^2$ shifted up 3 units: $g(x) = x^2 + 3$

Stretches and Shrinks

Vertical and Horizontal Stretches/Shrinks

Stretches and shrinks occur when a constant is multiplied inside or outside the function.

• Vertical stretch/shrink: $a f(x)$ stretches the graph vertically if $|a| > 1$ and shrinks if $0 < |a| < 1$.

• Horizontal stretch/shrink: $f(bx)$ shrinks the graph horizontally if $|b| > 1$ and stretches if $0 < |b| < 1$.

Type	Transformation	Effect
Vertical Stretch	$y = a f(x)$, $|a| > 1$	Stretches vertically
Vertical Shrink	$y = a f(x)$, $0 < |a| < 1$	Shrinks vertically
Horizontal Stretch	$y = f(bx)$, $0 < |b| < 1$	Stretches horizontally
Horizontal Shrink	$y = f(bx)$, $|b| > 1$	Shrinks horizontally

Example: $f(x) = x^2$, $g(x) = 2f(x) = 2x^2$ is a vertical stretch by a factor of 2.

Combining Transformations

Order of Transformations

Multiple transformations can be applied to a function. The typical order is:

1. Reflections

2. Stretches/Shrinks

3. Shifts

Example: $g(x) = -2f(x + 3) - 1$ means reflect over x-axis, stretch vertically by 2, shift left by 3, and down by 1.

Domain and Range of Transformed Functions

Effect of Transformations on Domain and Range

Transformations can change the domain and range of a function. To find the new domain and range, observe the effect of each transformation on the original graph.

• Vertical shifts affect the range but not the domain.

• Horizontal shifts affect the domain but not the range.

• Reflections may change the sign of the range or domain.

Original Function	Domain	Range
$f(x)$	$[a, b]$	$[c, d]$
$f(x) + k$	$[a, b]$	$[c + k, d + k]$
$f(x - h)$	$[a + h, b + h]$	$[c, d]$

Example: If $f(x)$ has domain $[-1, 1]$ and range $[0, 2]$, then $g(x) = f(x - 2) + 3$ has domain $[1, 3]$ and range $[3, 5]$.

Practice Problems and Applications

• Given $f(x) = x^2$, sketch $g(x) = -f(x - 2) + 1$.

• Find the equation for $g(x)$ if $g(x)$ is $f(x)$ reflected over the y-axis and shifted up 2 units.

• Given the graph of $f(x)$, determine the domain and range of $g(x) = f(x + 1) - 3$.

Additional info: These notes cover the essential transformations of functions, including graphical and algebraic representations, as well as the impact on domain and range. Practice problems are included to reinforce understanding.