Transformations of Functions: Precalculus Study Guide

Chapter 1: Graphs and Functions

1.7 Transformations of Functions

Transformations are operations that alter the graph of a function in predictable ways. Understanding these transformations is essential for graphing and analyzing functions in precalculus. The main types of transformations include shifts, reflections, stretches, and compressions.

Definition of Transformations

• Transformation: If a new function is formed by performing certain operations on a given function f, then the graph of the new function is called a transformation of the graph of f.

Vertical Shifts

Vertical shifts move the graph of a function up or down without changing its shape.

• Upward Shift: The graph of g(x) = f(x) + d is the graph of y = f(x) shifted d units up.

• Downward Shift: The graph of h(x) = f(x) - d is the graph of y = f(x) shifted d units down.

• Correspondence: If (x, y) is a point on y = f(x), then (x, y + d) is on g(x) = f(x) + d and (x, y - d) is on h(x) = f(x) - d.

Example: For f(x) = |x|, g(x) = |x| + 2, and h(x) = |x| - 3, the graphs are shifted vertically.

Horizontal Shifts

Horizontal shifts move the graph left or right.

• Right Shift: The graph of y = f(x - c) is the graph of y = f(x) shifted c units to the right.

• Left Shift: The graph of y = f(x + c) is the graph of y = f(x) shifted c units to the left.

• Correspondence: If (x, y) is a point on y = f(x), then (x + c, y) is on y = f(x - c) and (x - c, y) is on y = f(x + c).

Example: For f(x) = x^2, g(x) = (x - 2)^2, and h(x) = (x + 3)^2, the graphs are shifted horizontally.

Combined Vertical and Horizontal Shifts

Functions can be shifted both vertically and horizontally by combining the above transformations.

• General Form: y = f(x ± c) ± d shifts the graph horizontally by c units and vertically by d units.

• Domain and Range: Shifts may affect the domain and range of the function.

Example: Shifting y = \sqrt{x} horizontally and vertically.

Reflections

Reflections flip the graph over a specified axis.

• Reflection about the x-axis: The graph of g(x) = -f(x) is a reflection of y = f(x) about the x-axis. If (x, y) is on f, then (x, -y) is on g.

• Reflection about the y-axis: The graph of g(x) = f(-x) is a reflection of y = f(x) about the y-axis. If (x, y) is on f, then (-x, y) is on g.

Combining Transformations

Multiple transformations can be applied sequentially to a function. The order of operations matters and affects the final graph.

• Example: To graph y = -|x - 2| + 3 from y = |x|:

  1. Shift right 2 units: y = |x - 2|

  2. Reflect about x-axis: y = -|x - 2|

  3. Shift up 3 units: y = -|x - 2| + 3

Absolute Value Transformations

Taking the absolute value of a function reflects any portion below the x-axis above the x-axis.

• Example: For f(x) = (x + 1)^2 - 4, the graph of y = |f(x)| is unchanged for y \geq 0 and reflected for y < 0.

Vertical Stretching and Compressing

Vertical stretching or compressing changes the steepness of the graph by multiplying the y-coordinates by a constant.

• Vertical Stretch: If a > 1, g(x) = a f(x) stretches the graph away from the x-axis.

• Vertical Compression: If 0 < a < 1, g(x) = a f(x) compresses the graph toward the x-axis.

• Reflection: If a < 0, first stretch/compress, then reflect about the x-axis.

Example: For f(x) = |x|, g(x) = 2|x| (stretch), h(x) = \frac{1}{2}|x| (compression).

Horizontal Stretching and Compressing

Horizontal stretching or compressing changes the width of the graph by multiplying the x-coordinates by a constant.

• Horizontal Stretch: If 0 < b < 1, g(x) = f(bx) stretches the graph away from the y-axis.

• Horizontal Compression: If b > 1, g(x) = f(bx) compresses the graph toward the y-axis.

• Reflection: If b < 0, first stretch/compress, then reflect about the y-axis.

Multiple Transformations Example

Applying several transformations in sequence can produce complex graphs.

• Example: For f(x) = -2(x - 1)^2 + 3:

  1. Start with y = x^2 (basic function).

  2. Shift right 1 unit: y = (x - 1)^2.

  3. Stretch vertically by 2: y = 2(x - 1)^2.

  4. Reflect about x-axis: y = -2(x - 1)^2.

  5. Shift up 3 units: y = -2(x - 1)^2 + 3.

Application: Poiseuille’s Law for Arterial Blood Flow

Transformations are used in modeling real-world phenomena. Poiseuille’s Law describes blood flow velocity in an artery as a function of distance from the center.

• Formula: v(r) = c(R^2 - r^2), where c is a constant, R is the artery radius, and r is the distance from the center.

• Transformations: Start with y = r^2, stretch vertically, reflect about x-axis, and shift up.

• Physical Meaning: Velocity decreases from the center to the wall, ceasing at r = R.

Summary Table: Types of Transformations

Transformation	Equation	Effect
Vertical Shift	$y = f(x) + d$	Up $d$ units
Vertical Shift	$y = f(x) - d$	Down $d$ units
Horizontal Shift	$y = f(x - c)$	Right $c$ units
Horizontal Shift	$y = f(x + c)$	Left $c$ units
Reflection x-axis	$y = -f(x)$	Flip over x-axis
Reflection y-axis	$y = f(-x)$	Flip over y-axis
Vertical Stretch	$y = a f(x)$	Stretch if $a > 1$, compress if $0 < a < 1$
Horizontal Stretch	$y = f(bx)$	Stretch if $0 < b < 1$, compress if $b > 1$