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Unit Review: Functions and Quadratic Equations, Factored Form: a(x - r₁)(x…
Unit Review: Functions and Quadratic Equations
Functions and Relations
Definition of Function: A relation where each input has exactly one output.
Relationship: A function is a specific type of relation where each input (x) maps to only one output (y).
Definition of Relation: A set of ordered pairs (x, y).
Standard Form: a(x - p)² + q
Parabola: A U-shaped curve representing the graph of a quadratic function.
Vertex: Maximum/Minimum Point - The highest or lowest point on the parabola.
Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves. It passes through the vertex.
Domain: The set of all possible x-values. For quadratic functions, the domain is typically all real numbers (-∞, ∞).
Domain: The set of all possible x-values. For quadratic functions, the domain is typically all real numbers (-∞, ∞).
Factored Form: a(x - r₁)(x - r₂)
x-intercepts: r₁ and r₂
Effect of 'a'
Axis of Symmetry: x = (r₁ + r₂)/2
Identifying a, p, q: 'a' is the same as in standard form. 'p' and 'q' are the x-intercepts (roots) of the quadratic equation.
Finding x-intercepts: The x-intercepts are p and q.
Key Features of Quadratic Function
Parabola: Shape and Orientation
Vertex: Maximum/Minimum Point
Axis of Symmetry
Domain
Range
Factored Form: a(x - r₁)(x - r₂)
General Form: ax² + bx + c
Parabola: Shape and Orientation
a > 0: Parabola opens upwards (minimum point)
a < 0: Parabola opens downwards (maximum point)
Vertex: (h, k)
Axis of Symmetry: x = h
Domain: All real numbers
Range: Depends on parabola orientation