Algbra Unit 9

Absolutely — here’s the master version of Unit 9, fully merging List A and List B, with nothing left out. It’s your one-stop, Knowt-friendly masterlist:

Unit 9 – Masterlist: Parent Functions, Attributes, Transformations, and Solving

1. Parent Functions Overview (with Equations + Graph Shape)

• Linear: y = x
  Diagonal line, passes through origin
  Domain/Range: all real numbers
  Inverse: itself

• Quadratic: y = x²
  U-shape, vertex at (0, 0)
  Range: [0, ∞), domain: all real
  Inverse: square root (restricted)

• Absolute Value: y = |x|
  V-shape, vertex at (0, 0)
  Range: [0, ∞), domain: all real
  Inverse: piecewise

• Square Root: y = √x
  Curve starting at (0, 0), moves right
  Domain/Range: [0, ∞)
  Inverse: quadratic (restricted)

• Cube Root: y = ∛x
  S-curve through origin
  Domain/Range: all real
  Inverse: cubic

• Cubic: y = x³
  S-curve (steep), through origin
  Domain/Range: all real
  Inverse: cube root

• Quartic: y = x⁴
  Tight U-shape, even degree
  Domain: all real, Range: [0, ∞)
  Inverse: not a function without restriction

• Quintic: y = x⁵
  Steep S-curve, odd degree
  Domain/Range: all real
  Inverse: fifth root

• Rational: y = 1/x
  Two branches, asymptotes at x = 0 and y = 0
  Domain/Range: all reals except 0
  Inverse: itself

• Exponential Growth: y = b^x (b > 1)
  Flat then fast increase, asymptote y = 0
  Domain: all real, Range: (0, ∞)
  Inverse: log(x)

• Exponential Decay: y = (1/b)^x
  Decreasing curve, asymptote y = 0
  Same domain/range as growth

• Logarithmic Growth: y = log_b(x)
  Slow increase, asymptote x = 0
  Domain: (0, ∞), Range: all real
  Inverse: exponential

• Logarithmic Decay: y = log_{1/b}(x)
  Decreases slowly, vertical asymptote at x = 0
  Same domain/range as above

2. (h, k) Behavior by Function

• Vertex at (h, k):
  

  • Quadratic

  • Absolute Value

• Start Point at (h, k):
  

  • Square Root

• Inflection point at (h, k):
  

  • Cubic

  • Cube Root

• (h, k) doesn’t exist as a point:
  

  • Rational, Logarithmic

• Default (h, k) = (0, 0):
  

  • All except Rational and Logarithmic unless shifted

3. Intercepts

• Always 1 y-intercept:
  

  • Linear

  • Quadratic

  • Absolute Value

  • Exponential

  • Logarithmic

• May have no intercepts:
  

  • Rational

  • Exponential (no x-int if unshifted)

  • Square Root (if shifted)

• x-intercepts (typical counts):
  

  • Quadratic: 0, 1, or 2

  • Absolute Value: 0 or 1

  • Exponential: 0 or 1 (sometimes none)

  • Cube Root: always 1

  • Logarithmic: always 1

  • Square Root: 1 or none

4. Asymptotes

• Horizontal asymptote:
  

  • Exponential (y = 0)

  • Rational (depends on degrees)

• Vertical asymptote:
  

  • Rational (x = 0)

  • Logarithmic (x = 0)

• Both:
  

  • Rational (e.g., y = 1/x)

5. Domain & Range

• Domain: all real numbers
  

  • Linear

  • Cubic

  • Cube Root

  • Exponential

  • Logarithmic

• Range: all real numbers
  

  • Linear

  • Cubic

  • Cube Root

  • Logarithmic

6. End Behavior

• Linear:
  

  • x → ±∞, y → ±∞ (same direction as slope)

• Quadratic:
  

  • x → ±∞, y → ∞ (opens up), or y → –∞ (opens down)

• Cubic & Cube Root:
  

  • x → –∞, y → –∞

  • x → ∞, y → ∞

• Exponential Growth:
  

  • x → –∞, y → 0

  • x → ∞, y → ∞

• Exponential Decay:
  

  • x → –∞, y → ∞

  • x → ∞, y → 0

• Rational:
  

  • Approaches asymptotes on both ends

7. Transformations Recap

• Vertical shifts: f(x) ± k

• Horizontal shifts: f(x ± h)

• Vertical stretch/compression: a·f(x)

• Horizontal stretch/compression: f(bx)

• Reflections:
  

  • Over x-axis → –f(x)

  • Over y-axis → f(–x)

• Stretch = a > 1

• Compression = 0 < a < 1

8. Solving Strategies by Function

• Linear: Isolate x

• Quadratic: Factor, square root method, complete square, quadratic formula

• Absolute Value: Split into two equations

• Square Root: Isolate radical, square both sides

• Rational: Get common denominator or cross-multiply, check for restrictions

• Exponential: Use logarithms

• Logarithmic: Convert to exponential

• Cube Root & Cubic: Isolate variable, cube both sides or factor

9. Solution Types

• Extraneous solutions:
  

  • Rational

  • Square Root

  • Logarithmic

• Imaginary solutions:
  

  • Quadratic (when discriminant is negative)

• Exactly one real solution:
  

  • Linear

  • Cube Root

  • Exponential (usually)

• No solution (common cases):
  

  • Absolute value = negative

  • log of negative or 0

  • square root of negative number

10. Inverse Functions

• Linear ↔ Linear

• Quadratic ↔ Square Root (if restricted)

• Cubic ↔ Cube Root

• Exponential ↔ Logarithmic

• Square Root ↔ Quadratic (positive side only)

11. Piecewise Functions (Bonus Concept)

• Definition: A function defined by different rules over different parts of the domain

• Examples: Absolute value as a piecewise, step functions

• Key Skills: Know how to evaluate and graph piecewise by intervals