Transformations of Functions: Reflections, Shifts, and Stretches

Transformations of Functions

Introduction to Transformations

Transformations occur when a function is manipulated and changes position and/or shape on the coordinate plane. The main types of transformations include reflection, shift (translation), and stretch (scaling). Understanding these transformations is essential for analyzing and graphing functions in College Algebra.

Types of Function Transformations

Reflections

A reflection is a transformation where the function appears to be "folded" over a specific axis. Reflections can occur over the x-axis or y-axis, changing the sign of the output or input, respectively.

• Reflection over the x-axis: The output values (y-values) change sign. The transformation is given by: $f(x) \rightarrow -f(x)$

• Reflection over the y-axis: The input values (x-values) change sign. The transformation is given by: $f(x) \rightarrow f(-x)$

Example: If $f(x) = x^2$, then reflecting over the x-axis gives $g(x) = -x^2$; reflecting over the y-axis gives $h(x) = (-x)^2 = x^2$ (which is the same as the original for even functions).

Type of Reflection	Transformation	Effect on Graph
Over x-axis	$-f(x)$	Flips graph upside down
Over y-axis	$f(-x)$	Flips graph left-right

Shifts (Translations)

A shift occurs when a function is moved vertically and/or horizontally from its original position. Shifts do not change the shape of the graph, only its location.

• Vertical Shift: Adding or subtracting a constant $k$ to the function moves the graph up or down. General form: $f(x) + k$ If $k > 0$, shift up; if $k < 0$, shift down.

• Horizontal Shift: Adding or subtracting a constant $h$ inside the function argument moves the graph left or right. General form: $f(x - h)$ If $h > 0$, shift right; if $h < 0$, shift left.

Example: $f(x) = x^2$ shifted right by 3 and up by 2: $g(x) = (x - 3)^2 + 2$

Type of Shift	Transformation	Effect on Graph
Vertical (up/down)	$f(x) + k$	Up if $k > 0$, down if $k < 0$
Horizontal (left/right)	$f(x - h)$	Right if $h > 0$, left if $h < 0$

Stretches and Compressions

A stretch or compression changes the shape of the graph by making it narrower or wider. This is done by multiplying the function by a constant $c$.

• Vertical Stretch/Compression: $c \cdot f(x)$

  • If $|c| > 1$, the graph is stretched (narrower).

  • If $0 < |c| < 1$, the graph is compressed (wider).

Example: $f(x) = x^2$; $g(x) = 2x^2$ is a vertical stretch by 2; $h(x) = 0.5x^2$ is a vertical compression by 0.5.

Type	Transformation	Effect on Graph
Vertical Stretch	$c f(x)$, $|c| > 1$	Narrower
Vertical Compression	$c f(x)$, $0 < |c| < 1$	Wider

Combining Transformations

Multiple transformations can be applied to a function in sequence. The order of operations is important: typically, stretches/compressions and reflections are applied before shifts.

• General Form: $y = a f(b(x - h)) + k$

  • $a$ = vertical stretch/compression and/or reflection over x-axis

  • $b$ = horizontal stretch/compression and/or reflection over y-axis

  • $h$ = horizontal shift

  • $k$ = vertical shift

Example: $g(x) = -|x - 3| + 2$ is the absolute value function reflected over the x-axis, shifted right by 3, and up by 2.

Worked Examples and Practice Problems

Example 1: Matching Transformations

Given $f(x) = |x|$, match the following functions to their graphs:

• $p(x) = |x - 3| + 2$ (shift right 3, up 2)

• $q(x) = -|x|$ (reflection over x-axis)

• $r(x) = -4|x|$ (reflection over x-axis, vertical stretch by 4)

Example 2: Reflection over the x-axis

Given $f(x) = x + 2$, the reflection over the x-axis is $h(x) = -x - 2$.

Example 3: Reflection over the y-axis

Given $f(x) = \sqrt{x - 2}$, the reflection over the y-axis is $g(x) = \sqrt{-x - 2}$.

Example 4: Shifting a Function

Given $f(x)$, sketch the transformation $f(x - 2) + 3$ (shift right 2, up 3).

Example 5: Combining Reflections and Shifts

If $h(x)$ is a transformation of $f(x) = x^3$ reflected over the x-axis and shifted down 2 units, then $h(x) = -x^3 - 2$.

Summary Table: Transformations of Functions

Transformation	Equation	Description
Reflection over x-axis	$-f(x)$	Flips graph upside down
Reflection over y-axis	$f(-x)$	Flips graph left-right
Vertical shift	$f(x) + k$	Up if $k > 0$, down if $k < 0$
Horizontal shift	$f(x - h)$	Right if $h > 0$, left if $h < 0$
Vertical stretch	$c f(x)$, $|c| > 1$	Narrower
Vertical compression	$c f(x)$, $0 < |c| < 1$	Wider

Key Takeaways

• Transformations allow you to predict and sketch the graph of a function after it has been manipulated.

• Reflections, shifts, and stretches/compressions can be combined for more complex transformations.

• Understanding the algebraic form of each transformation helps in graphing and analyzing functions efficiently.