Transformations and Parent Functions

Transformations

• Vertical Stretch:
  • Example: 2f(x) means the graph will be twice as large vertically.
• Vertical Shrink:
  • Example: 1/2f(x) means the graph will be shorter vertically but the same width.
• Horizontal Shrink:
  • Example: f(2x) means the period is halved (e.g., from 2π to π), same height.
• Horizontal Stretch:
  • Example: f(1/2x) means the period doubles (e.g., from 2π to 4π), same height.
• Vertical Shift:
  • f(x)+2 shifts up 2 units.
  • f(x)-2 shifts down 2 units.
• Horizontal Shift:
  • f(x-2) shifts right 2 units.
  • f(x+2) shifts left 2 units.

Reflections

• Reflection over x-axis: -f(x) changes positive y-values to negative.
• Reflection over y-axis: f(-x) mirrors the function over y-axis.
• Reflection over origin: Combining both above transformations.

Inverse and Reciprocal Functions

• Inverse Function: Reflects across y=x.
• Reciprocal Function: 1/f(x); an increasing function becomes decreasing.

Graphs of Parent Functions

Linear Function

• Equation: y = x
• Domain/Range: All real numbers
• End Behavior: As x → ±∞, y → ±∞

Quadratic Function

• Equation: y = x^2
• Domain: All real numbers
• Range: [0, ∞)
• End Behavior: As x → ±∞, y → ∞](streamdown:incomplete-link)

Cubic Function

• Equation: y = x^3
• Domain/Range: All real numbers
• End Behavior: As x → ±∞, y → ±∞

Absolute Value Function

• Equation: y = |x|
• Domain: All real numbers
• Range: [0, ∞)](streamdown:incomplete-link)

Square Root Function

• Equation: y = √x
• Domain/Range: [0, ∞)](streamdown:incomplete-link)

Rational Function

• Equation: y = 1/x
• Domain: (-∞, 0) ∪ (0, ∞)
• Range: (-∞, 0) ∪ (0, ∞)
• Asymptotes: Vertical at x = 0, horizontal at y = 0

Exponential Function

• Equation: y = e^x
• Domain: All real numbers
• Range: (0, ∞)
• Asymptote: Horizontal at y = 0

Logarithmic Function

• Equation: y = log_b(x)
• Domain: (0, ∞)
• Range: All real numbers

Trigonometric Functions

Sine and Cosine

• Domain: All real numbers
• Range: [-1, 1]
• Amplitude: Distance from center to peak

Tangent

• Domain: Excludes nπ/2, n odd integers
• Range: All real numbers

Cosecant

• Domain: Excludes multiples of π
• Range: (-∞, -1] ∪ [1, ∞)](streamdown:incomplete-link)

Inverse Trigonometric Functions

Inverse Sine

• Domain: [-1, 1]
• Range: [-π/2, π/2]

Inverse Cosine

• Domain: [-1, 1]
• Range: [0, π]

Inverse Tangent

• Domain: All real numbers
• Range: (-π/2, π/2)
• Behavior: Approaches horizontal asymptotes at ±π/2 as x approaches ±∞