Trigonometry Study Guide: Functions, Transformations, Symmetry, and Complex Numbers

2.5 Parent Functions – Horizontal and Vertical Shifts

Introduction to Parent Functions and Shifts

Parent functions are the simplest forms of functions in a family, such as linear, quadratic, cubic, and trigonometric functions. Understanding how these functions shift horizontally and vertically is essential for graphing and analyzing their behavior.

• Parent Function: The basic form of a function before any transformations are applied (e.g., $f(x) = x$, $f(x) = x^2$, $f(x) = \sin x$).

• Horizontal Shift: Moving the graph left or right. For $f(x)$, $f(x - h)$ shifts right by $h$ units, $f(x + h)$ shifts left by $h$ units.

• Vertical Shift: Moving the graph up or down. For $f(x)$, $f(x) + k$ shifts up by $k$ units, $f(x) - k$ shifts down by $k$ units.

• General Transformation Formula: $f(x) \rightarrow f(x - h) + k$

Example: The graph of $f(x) = x^2$ shifted 3 units right and 2 units up is $f(x) = (x - 3)^2 + 2$.

2.5 Transformations – Reflections and Stretches

Reflections and Stretches of Functions

Transformations alter the appearance of a function's graph. Reflections flip the graph over an axis, while stretches change its steepness or width.

• Reflection over x-axis: $y = -f(x)$

• Reflection over y-axis: $y = f(-x)$

• Vertical Stretch/Compression: $y = a f(x)$, where $|a| > 1$ stretches, $0 < |a| < 1$ compresses.

• Horizontal Stretch/Compression: $y = f(bx)$, where $|b| > 1$ compresses, $0 < |b| < 1$ stretches horizontally.

• Domain and Range (D/R): State the set of possible input (domain) and output (range) values for each transformation.

Example: $f(x) = x^2$ reflected over the x-axis and stretched vertically by 2: $f(x) = -2x^2$.

2.4 Symmetry

Types of Symmetry in Functions

Symmetry helps classify functions and predict their behavior. The most common types are even, odd, and origin symmetry.

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain. Graph is symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain. Graph is symmetric about the origin.

• No Symmetry: Functions that do not satisfy either condition.

Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

2.3 Composite Functions

Combining Functions

Composite functions involve applying one function to the result of another. This is denoted as $(f \circ g)(x) = f(g(x))$.

• Notation: $(f \circ g)(x) = f(g(x))$

• Order Matters: $(f \circ g)(x) \neq (g \circ f)(x)$ in general.

• Domain: The domain of the composite function is restricted by both $g(x)$ and $f(x)$.

Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = f(g(x)) = (x + 1)^2$.

5.1 Inverses

Inverse Functions

An inverse function reverses the effect of the original function. If $f(x)$ maps $x$ to $y$, then $f^{-1}(y)$ maps $y$ back to $x$.

• Definition: $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$.

• Finding Inverses: Solve $y = f(x)$ for $x$ in terms of $y$, then swap $x$ and $y$.

• Graphical Relationship: The graph of $f^{-1}(x)$ is the reflection of $f(x)$ over the line $y = x$.

Example: $f(x) = 2x + 3$; solve $y = 2x + 3$ for $x$: $x = \frac{y - 3}{2}$, so $f^{-1}(x) = \frac{x - 3}{2}$.

Review

Summary and Practice

Review covers all previous topics, focusing on understanding transformations, symmetry, composite functions, and inverses. Practice problems reinforce these concepts.

• Key Skills: Identifying and applying transformations, determining symmetry, composing functions, and finding inverses.

• Explanation: Be able to explain each concept and solve related problems.

Complex Numbers

Introduction to Complex Numbers

Complex numbers extend the real number system to include solutions to equations like $x^2 + 1 = 0$. They are written in the form $a + bi$, where $i$ is the imaginary unit.

• Imaginary Unit: $i$ is defined by $i^2 = -1$.

• Complex Number: $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

• Operations: Addition, subtraction, multiplication, and division follow algebraic rules, with $i^2 = -1$.

Example: $(2 + 3i) + (1 - 4i) = 3 - i$

Test

Assessment of Understanding

The test covers all topics listed above, including parent functions, transformations, symmetry, composite functions, inverses, and complex numbers. Students should be prepared to solve problems and explain concepts.

Assignment and Topic Table

Overview of Assignments and Topics

Assignment	Topic	Homework
1	2.5 Parent Functions – horizontal and vertical shifts	Pg 139 #3, 6, 7, 8, 15, 26, 33, 38, 41, 47, 50, 51
2	2.5 Transformations – Reflections and stretches	Pg 139 #37, 40, 42, 43, 45, 46, 48, [49, 53, 55, state D/R also], 59, 62, 64
3	2.4 Symmetry	p. 139: #39, 44, 52, 57, 65, 66, 91, 92; p. 127: #2 -6 even, 7, 8, 35, 39, 40
4	2.3 Composite Functions 5.1 Inverses	p. 128: #41, 55; p. 120: #1, 5, 23, 25, 27, 31; p. 315: #1, 25, 28, 35, 37, 49 (no calc.) 57, 62, 63, 67, 77, 79, 97
5	Review Complex Numbers	p. 129: #56, 57a, 58, 59 (explain); p. 158: #18, 51-59 odd, 61-67; worksheet
6	Test	Day 1 of next unit; p. 315: #1, 25, 28, 35, 37, 49 (no calc.) 57, 62, 63, 67, 77, 79, 97