1. PROPERTIES OF GRAPHS AND FUNCTIONS
1.1 Interpret a Function from Multiple Perspectives
A Function is a relationship where each input (x) has exactly one output (y).
Four Perspectives:
Perspective Description Example
Algebraic Formula/equation f(x) = 2x + 3
Numerical Table of values x: 1, 2, 3 → y: 5, 7, 9
Graphical Visual plot on coordinate plane Line passing through points
Verbal Word description "Double the input and add 3"
Extracting Information:
• Input-output pairs: From table or graph
• Pattern: How y changes with x
• Behavior: Increasing, decreasing, constant
• Special points: Intercepts, maximum, minimum
1.2 Find the Domain and Range of a Function
DOMAIN: All possible input values (x-values)
RANGE: All possible output values (y-values)
How to Find:
Function Type Domain Range
Linear f(x) = mx + b All real numbers: (-∞, ∞) All real numbers: (-∞, ∞)
Quadratic f(x) = ax² + bx + c All real numbers: (-∞, ∞) y ≥ minimum (if a > 0) or y ≤ maximum (if a < 0)
,Function Type Domain Range
Square Root f(x) = √x x ≥ 0:
Rational f(x) = 1/x x ≠ 0: (-∞, 0) ∪ (0, ∞) y ≠ 0: (-∞, 0) ∪ (0, ∞)
Exponential f(x) = eˣ All real numbers: (-∞, ∞) y > 0: (0, ∞)
Logarithmic f(x) = ln(x) x > 0: (0, ∞) All real numbers: (-∞, ∞)
Steps to Find Domain:
1. Look for denominators (set ≠ 0)
2. Look for square roots (set ≥ 0)
3. Look for logarithms (set > 0)
4. Otherwise, domain is all real numbers
Steps to Find Range:
1. Graph the function or analyze behavior
2. Find minimum/maximum values
3. Determine all possible y-values
1.3 Identify the Graph of a Function Using Vertical Line Test
Vertical Line Test:
• Draw any vertical line through the graph
• If the line touches the graph at most once → it's a function
• If the line touches the graph more than once → NOT a function
Why it works:
• A function must have exactly ONE output for each input
• Vertical line = one x-value
• Multiple intersections = multiple y-values for same x → not a function
Examples:
, • Circle x² + y² = r² → NOT a function (vertical line hits twice)
• Parabola y = x² → IS a function (vertical line hits once)
• Line y = 2x + 1 → IS a function
1.4 Identify Symmetries of Graphs
Types of Symmetry:
Symmetry Type Definition Test Example
y-axis (Even) Symmetric about y-axis f(-x) = f(x) f(x) = x²
Origin (Odd) Symmetric about origin f(-x) = -f(x) f(x) = x³
x-axis Symmetric about x-axis Replace y with -y Not a function
Even Functions (y-axis symmetry):
• f(x) = x², x⁴, x⁶ (even powers)
• f(x) = cos(x)
• Graph looks same on left and right of y-axis
Odd Functions (origin symmetry):
• f(x) = x³, x⁵, x⁷ (odd powers)
• f(x) = sin(x)
• Graph rotates 180° about origin
How to Test:
1. Replace x with -x
2. Simplify
3. Compare to original f(x) or -f(x)
1.5 Different Types of Functions and Their Properties
1.1 Interpret a Function from Multiple Perspectives
A Function is a relationship where each input (x) has exactly one output (y).
Four Perspectives:
Perspective Description Example
Algebraic Formula/equation f(x) = 2x + 3
Numerical Table of values x: 1, 2, 3 → y: 5, 7, 9
Graphical Visual plot on coordinate plane Line passing through points
Verbal Word description "Double the input and add 3"
Extracting Information:
• Input-output pairs: From table or graph
• Pattern: How y changes with x
• Behavior: Increasing, decreasing, constant
• Special points: Intercepts, maximum, minimum
1.2 Find the Domain and Range of a Function
DOMAIN: All possible input values (x-values)
RANGE: All possible output values (y-values)
How to Find:
Function Type Domain Range
Linear f(x) = mx + b All real numbers: (-∞, ∞) All real numbers: (-∞, ∞)
Quadratic f(x) = ax² + bx + c All real numbers: (-∞, ∞) y ≥ minimum (if a > 0) or y ≤ maximum (if a < 0)
,Function Type Domain Range
Square Root f(x) = √x x ≥ 0:
Rational f(x) = 1/x x ≠ 0: (-∞, 0) ∪ (0, ∞) y ≠ 0: (-∞, 0) ∪ (0, ∞)
Exponential f(x) = eˣ All real numbers: (-∞, ∞) y > 0: (0, ∞)
Logarithmic f(x) = ln(x) x > 0: (0, ∞) All real numbers: (-∞, ∞)
Steps to Find Domain:
1. Look for denominators (set ≠ 0)
2. Look for square roots (set ≥ 0)
3. Look for logarithms (set > 0)
4. Otherwise, domain is all real numbers
Steps to Find Range:
1. Graph the function or analyze behavior
2. Find minimum/maximum values
3. Determine all possible y-values
1.3 Identify the Graph of a Function Using Vertical Line Test
Vertical Line Test:
• Draw any vertical line through the graph
• If the line touches the graph at most once → it's a function
• If the line touches the graph more than once → NOT a function
Why it works:
• A function must have exactly ONE output for each input
• Vertical line = one x-value
• Multiple intersections = multiple y-values for same x → not a function
Examples:
, • Circle x² + y² = r² → NOT a function (vertical line hits twice)
• Parabola y = x² → IS a function (vertical line hits once)
• Line y = 2x + 1 → IS a function
1.4 Identify Symmetries of Graphs
Types of Symmetry:
Symmetry Type Definition Test Example
y-axis (Even) Symmetric about y-axis f(-x) = f(x) f(x) = x²
Origin (Odd) Symmetric about origin f(-x) = -f(x) f(x) = x³
x-axis Symmetric about x-axis Replace y with -y Not a function
Even Functions (y-axis symmetry):
• f(x) = x², x⁴, x⁶ (even powers)
• f(x) = cos(x)
• Graph looks same on left and right of y-axis
Odd Functions (origin symmetry):
• f(x) = x³, x⁵, x⁷ (odd powers)
• f(x) = sin(x)
• Graph rotates 180° about origin
How to Test:
1. Replace x with -x
2. Simplify
3. Compare to original f(x) or -f(x)
1.5 Different Types of Functions and Their Properties
