Precalculus Study Guide: Function Composition, Transformations, and Quadratic Functions

Section 5.1: Function Composition

Definition and Evaluation of Function Composition

Function composition involves applying one function to the results of another. If you have two functions, f(x) and g(x), the composition is written as f(g(x)) or (f ˆ g)(x).

• Definition: The composition f(g(x)) means you first apply g to x, then apply f to the result.

• Evaluation Example: If f(x) = 2x + 1 and g(x) = x^2, then f(g(x)) = f(x^2) = 2x^2 + 1.

• Explicit Formula: Given f(x) and g(x), substitute g(x) into every instance of x in f(x) to find f(g(x)).

Input and Output Units in Function Composition

When composing functions, the output units of the inner function must match the input units of the outer function.

• Example: If g(x) outputs time in seconds and f(x) takes time in seconds as input, then f(g(x)) is valid.

• Unit Consistency: Always check that the units "line up" when composing functions.

Order of Composition

The order in which you compose functions matters: f(g(x)) is generally not the same as g(f(x)).

• Example: If f(x) = x + 1 and g(x) = 2x, then f(g(x)) = 2x + 1 but g(f(x)) = 2(x + 1) = 2x + 2.

Domain of Composite Functions

The domain of f(g(x)) consists of all x in the domain of g such that g(x) is in the domain of f.

• Example: If f(x) = \sqrt{x} and g(x) = x - 2, then f(g(x)) = \sqrt{x - 2} is defined only for x \geq 2.

Section 5.2 - 5.6: Function Transformations

Standard Form and Types of Transformations

The standard form for a function transformation is y = a f(bx + h) + k, where each parameter affects the graph in a specific way.

• a: Vertical stretch (|a| > 1) or compression (0 < |a| < 1); negative a reflects over the x-axis.

• b: Horizontal stretch/compression and reflection over the y-axis if negative.

• h: Horizontal shift (left if h > 0, right if h < 0).

• k: Vertical shift (up if k > 0, down if k < 0).

Effect of Transformations on Graphs

• Input Transformations: Affect the x-values (horizontal changes).

• Output Transformations: Affect the y-values (vertical changes).

• Order of Transformations: Generally, apply stretches/compressions and reflections before shifts.

• Example: For y = -2f(3x + 1) - 4, the graph is reflected over the x-axis, vertically stretched by 2, horizontally compressed by 1/3, shifted left by 1/3, and down by 4.

Inverse Transformations

Inverse transformations "undo" the effect of the original transformation. For example, if a function is shifted right by 2, the inverse shifts left by 2.

• Example: If f(x) = x^2 and g(x) = f(x - 3), then the inverse transformation is f(x + 3).

Piecewise and Absolute Value Transformations

• Piecewise Functions: Functions defined by different expressions over different intervals.

• Absolute Value Transformations: Applying absolute value to a function reflects all negative y-values above the x-axis.

Section 6.1 - 6.2: Quadratic Functions

Forms of Quadratic Functions

Quadratic functions can be written in three main forms:

• Standard Form: $y = ax^2 + bx + c$

• Factored Form: $y = a(x - r_1)(x - r_2)$

• Vertex Form: $y = a(x - h)^2 + k$

Converting Between Forms

• Standard to Vertex Form: Complete the square.

• Standard to Factored Form: Factor the quadratic, if possible.

• Factored to Standard Form: Expand the product.

Properties of Quadratic Functions

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Vertex: $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$

• Direction: Opens upward if a > 0, downward if a < 0.

• Maximum/Minimum: The vertex gives the maximum (if a < 0) or minimum (if a > 0) value.

Factored Form Insights

• Roots: The values r_1 and r_2 are the x-intercepts (if real).

• Existence: Not all quadratics can be factored over the real numbers (discriminant < 0).

• Example: $y = x^2 - 4 = (x - 2)(x + 2)$ has roots at x = 2 and x = -2.

Vertex Form Insights

• Vertex: The point (h, k) is the vertex of the parabola.

• Shifts: h shifts the graph horizontally, k shifts it vertically.

• Example: $y = 2(x - 1)^2 + 3$ has vertex at (1, 3).

Summary Table: Quadratic Forms and Their Properties

Form	Equation	Key Features
Standard	$y = ax^2 + bx + c$	Easy to expand; use for general properties
Factored	$y = a(x - r_1)(x - r_2)$	Shows x-intercepts (roots) directly
Vertex	$y = a(x - h)^2 + k$	Shows vertex and axis of symmetry directly

Graphing Quadratic Functions

• Identify the form of the quadratic to quickly find intercepts, vertex, and axis of symmetry.

• Plot the vertex and a few points on either side to sketch the parabola.

• Use Desmos or graphing calculators to check your work.

Examples

• Example 1: $y = x^2 - 6x + 8$ Factored form: $y = (x - 2)(x - 4)$; roots at x = 2, 4; vertex at $x = 3$, $y = -1$.

• Example 2: $y = 2(x + 1)^2 - 5$ Vertex at (-1, -5); opens upward; axis of symmetry $x = -1$.