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

2.5 Parent Functions – Horizontal and Vertical Shifts

Understanding Parent Functions and Their Shifts

Parent functions are the simplest forms of functions in various families, such as linear, quadratic, cubic, and absolute value functions. Shifting these functions horizontally or vertically changes their position on the coordinate plane without altering their basic shape.

• Parent Function: The most basic form of a function, e.g., $f(x) = x^2$ for quadratic functions.

• 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. $f(x) + k$ shifts up by $k$ units, $f(x) - k$ shifts down by $k$ units.

• General Form: $f(x) = a(x - h) + k$

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

2.5 Transformations – Reflections and Stretches

Transforming Parent Functions: Reflections and Stretches

Transformations alter the appearance of a function's graph. Reflections flip the graph over a line, while stretches and compressions change its size.

• 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 horizontally, $0 < |b| < 1$ stretches.

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

• Example: $y = -2|x|$ is a reflection of $y = |x|$ over the x-axis and vertically stretched by a factor of 2.

2.4 Symmetry

Identifying Symmetry in Functions

Symmetry in functions helps classify their graphs and predict their behavior. The main types are symmetry about the y-axis, x-axis, and origin.

• Even Function (y-axis symmetry): $f(-x) = f(x)$ for all $x$ in the domain.

• Odd Function (origin symmetry): $f(-x) = -f(x)$ for all $x$ in the domain.

• Neither: If neither condition holds, the function is neither even nor odd.

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

2.3 Composite Functions

Combining Functions: Composition

Composite functions are formed by applying one function to the result of another. The notation $(f \circ g)(x)$ means $f(g(x))$.

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

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

• Example: If $f(x) = 2x$ and $g(x) = x + 3$, then $(f \circ g)(x) = f(x + 3) = 2(x + 3) = 2x + 6$.

5.1 Inverses

Understanding Inverse Functions

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

• Definition: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

• Finding the Inverse: Swap $x$ and $y$ in $y = f(x)$, then solve for $y$.

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

• Example: $f(x) = 2x + 1$; to find $f^{-1}(x)$, set $y = 2x + 1$, swap to $x = 2y + 1$, solve: $y = \frac{x - 1}{2}$, so $f^{-1}(x) = \frac{x - 1}{2}$.

Review and Complex Numbers

Review of Key Concepts and Introduction to Complex Numbers

Reviewing previous topics consolidates understanding. Complex numbers extend the real number system to include solutions to equations like $x^2 + 1 = 0$.

• Complex Number: A number of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ($i^2 = -1$).

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

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

• Conjugate: The conjugate of $a + bi$ is $a - bi$.

Test Preparation

Preparing for the Unit Test

Review all previous topics, practice problems, and ensure understanding of key concepts such as function transformations, symmetry, composition, inverses, and complex numbers.

• Practice: Work through assigned problems and review mistakes.

• Conceptual Understanding: Be able to explain and apply each concept, not just perform calculations.

Summary Table of Assignments

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