Transformations of Functions: Precalculus Study Notes

Transformations of Functions

Overview of Function Transformations

Understanding how to transform functions is essential for graphing and analyzing mathematical models in precalculus. Transformations allow us to modify parent functions to create new functions by shifting, stretching, shrinking, or reflecting their graphs.

• Parent Function: The basic form of a function, such as y = f(x), before any transformations are applied.

• Key Points: Specific points on the parent function used to track changes during transformations.

Types of Transformations

There are four main types of transformations applied to functions:

• Horizontal Shifts: Move the graph left or right by adding or subtracting a value to the x-coordinate.

  • Left: $y = f(x + h)$

  • Right: $y = f(x - h)$

• Vertical Shifts: Move the graph up or down by adding or subtracting a value to the y-coordinate.

  • Up: $y = f(x) + k$

  • Down: $y = f(x) - k$

• Vertical Stretch/Shrink: Multiply the y-coordinates by a constant a.

  • Stretch: $a > 1$

  • Shrink: $0 < a < 1$

  • General form: $y = a f(x)$

• Reflection: Flip the graph over the x-axis or y-axis.

  • Over x-axis: $y = -f(x)$

  • Over y-axis: $y = f(-x)$

Transformation Chart

The following table summarizes the main transformations:

Transformation	Equation	Effect
Horizontal Shift Left	$y = f(x + h)$	Shift left by h units
Horizontal Shift Right	$y = f(x - h)$	Shift right by h units
Vertical Shift Up	$y = f(x) + k$	Shift up by k units
Vertical Shift Down	$y = f(x) - k$	Shift down by k units
Vertical Stretch	$y = a f(x)$, $a > 1$	Stretches graph vertically
Vertical Shrink	$y = a f(x)$, $0 < a < 1$	Shrinks graph vertically
Reflection over x-axis	$y = -f(x)$	Flips graph over x-axis
Reflection over y-axis	$y = f(-x)$	Flips graph over y-axis

Examples of Transformations

Let's apply transformations to specific functions and analyze their effects.

Example 1: Graphing Functions Using Transformations

• a. $f(x) = x - 2^3 - 3$

  • Apply horizontal shift, vertical shift, and possibly stretch/shrink.

  • Domain: All real numbers

  • Range: All real numbers

• b. $f(x) = 1 - x + 2$

  • Apply reflection, horizontal shift, and vertical shift.

  • Domain: All real numbers

  • Range: All real numbers

• c. $f(x) = -3x - 4 - 2$

  • Apply vertical stretch, reflection, horizontal and vertical shifts.

  • Domain: All real numbers

  • Range: All real numbers

• d. $f(x) = 1x + 2 - 1$

  • Apply vertical shift and horizontal shift.

  • Domain: All real numbers

  • Range: All real numbers

Example 2: Composite Transformations

Given the graph of $y = f(x)$, graph $g(x) = -12 f(x - 3) - 2$ by applying the following transformations:

• Horizontal shift right by 3 units

• Vertical stretch by factor of 12

• Reflection over x-axis

• Vertical shift down by 2 units

Example 3: Finding the Equation from a Transformed Graph

If $g$ is obtained from $f(x) = x^2$ through a sequence of transformations, and the graph shows an upward opening parabola with vertex at (1, 3), the equation for $g$ can be found by:

• Identifying the vertex form: $g(x) = a(x - h)^2 + k$

• Here, $h = 1$, $k = 3$

• Equation: $g(x) = (x - 1)^2 + 3$

Practice Problems

• 1. $f(x) = x^2 + 1$

  • Vertical shift up by 1 unit

  • Domain: All real numbers

  • Range: $y  1$

• 2. $f(x) = -2^3 x - 3$

  • Vertical stretch, reflection, and vertical shift

  • Domain: All real numbers

  • Range: All real numbers

Summary Table: Transformation Effects

Transformation	How to Apply	Result
Horizontal Shift	Add/subtract to x	Moves graph left/right
Vertical Shift	Add/subtract to y	Moves graph up/down
Stretch/Shrink	Multiply y by a	Stretches/shrinks vertically
Reflection	Change sign of x or y	Flips graph over axis

Additional info: The notes emphasize the importance of knowing parent functions and their key points for successful graphing. Practice problems reinforce the application of transformations and domain/range analysis.