Rucete ✏ AP Precalculus In a Nutshell
4. Parent Functions
This chapter introduces parent functions, how to graph them, transform them, compose functions, model real-life scenarios, and find inverses. These are essential foundations for function analysis throughout precalculus and beyond.
- Basic Parent Functions and Graphs
- Each parent function is the simplest form of its type (e.g., linear, quadratic, absolute value).
- Example functions:
- Linear: f(x) = x
- Quadratic: f(x) = x²
- Absolute value: f(x) = |x|
- Square root, cubic, cube root, rational, etc.
- Piecewise-Defined Functions
- Defined by multiple expressions over different intervals
- Graph boundaries must be analyzed for:
- Open/closed points
- Continuity or jump discontinuity
- Example: f(x) = { −x if x < 0, x if x ≥ 0 } → graph of |x|
- Function Transformations
- Vertical translations: f(x) + k shifts graph up/down
- Horizontal translations: f(x + h) shifts left/right
- Vertical dilation/reflection: af(x)
- a > 1 = stretch
- 0 < a < 1 = compress
- a < 0 = reflect
- Horizontal dilation/reflection: f(bx)
- b > 1 = compress
- 0 < b < 1 = stretch
- b < 0 = reflect
- Order of transformations:
- Horizontal shift
- Dilations
- Reflections
- Vertical shift
- Composite transformations: Combine multiple steps algebraically and geometrically
- Function Models and Applications
- Function types matched with scenarios:
- Linear → constant rate (e.g., rental pricing)
- Quadratic → symmetry, max/min (e.g., projectile motion)
- Cubic → volume or 3D modeling
- Polynomial → real zeros and turning points
- Piecewise → variable rates, taxes
- Rational → inverse relationships (e.g., gravity, work problems)
- Use diagrams and equations to restrict domain/range, and interpret intercepts, slopes, and variables
- Regression Models
- Use calculators to find best-fit equations from data:
- Linear regression (y = mx + b)
- Quadratic regression (y = ax² + bx + c)
- Interpretation includes:
- Slope: rate of change
- Intercepts: context-specific meanings
- Implied domain: practical limits of x-values
- Rational and Quadratic Modeling Examples
- Inverse variation: y = k/x
- Rational models appear in fencing, force, and cost problems
- Quadratic models used in optimization: area, height, volume problems
- Composition of Functions
- Composite function: f(g(x)) means plug g into f
- Order matters: f(g(x)) ≠ g(f(x))
- Use graphs, tables, equations, or verbal forms
- Identity function: f(x) = x satisfies f(f(x)) = x
- Domain of f(g(x)) must respect:
- Domain of g(x)
- Outputs of g(x) must lie within domain of f(x)
- Inverse Functions
- Inverse function f⁻¹ reverses input/output: f(a) = b ⇒ f⁻¹(b) = a
- To find f⁻¹:
- Swap x and y
- Solve for y
- Graphically: reflect across y = x
- Use horizontal line test to check for inverses
- One-to-one function: different inputs → different outputs
- Domain restriction may be needed (e.g., x² is not one-to-one unless x ≥ 0)
In a Nutshell
Parent functions are the core of understanding all function types. Knowing their basic forms and how to transform, compose, model, and invert them sets the foundation for real-world problem solving. Graphs and algebraic rules work hand-in-hand to reveal relationships, behaviors, and applications essential for success in precalculus.