Algebra Parent Functions Flashcards

Study Notes

Linear Function Parent: Defined by the equation f(x) = x, representing a straight line through the origin.
Quadratic Function Parent: Characterized by f(x) = x², resulting in a parabolic curve that opens upwards.
Absolute Value Function Parent: Expressed as f(x) = |x|, displaying a "V" shape graph, indicating distance from zero.
Square Root Function Parent: Given by f(x) = √x, featuring a graph that starts at the origin and increases gradually.
Cubic Function Parent: Defined by f(x) = x³, producing an S-shaped curve crossing through the origin.
Cube Root Function Parent: Given by f(x) = ∛x, which produces a curve allowing all real numbers.
Exponential Function Parent: Represented by f(x) = 2^x, revealing rapid growth for positive x values.
Reciprocal Function Parent: Defined by f(x) = 1/x, featuring two hyperbolas in opposite quadrants.

Linear Graphing Form: y - y₁ = m(x - x₁), representing a line on a graph using a point and a slope.
Quadratic Graphing Form: y = a(x - h)² + k, providing transformations through vertex form with 'a', 'h', and 'k'.
Absolute Value Graphing Form: y = a|x - h| + k, indicating transformations of the absolute value function.
Square Root Graphing Form: y = a√(x - h) + k, involving vertical/horizontal stretches and shifts of the square root function.
Cubic Graphing Form: y = a(x - h)³ + k, translating cubic functions while adjusting its steepness.
Cube Root Graphing Form: y = a∛(x - h) + k, similar transformations applicable as in other parent functions.

Function Definition: A relation connecting inputs to unique outputs, ensuring each input corresponds to only one output.
One-to-One Function: Each input maps to a unique output, and vice versa, ensuring no value is repeated in either domain or range.
Domain: Represents all possible x values that can be plugged into a function without resulting in undefined outputs.
Range: Encompasses all potential y values produced by the function as outputs across the domain.

Transformations: Graph alterations such as stretching, compressing, or reflecting, that retain the type of graph.
Absolute Value: Defines the numerical distance of any value from zero on the number line.
Parameter 'a': Indicates vertical stretch/shrink of the graph and shows reflection across the x-axis if negative.
Parameter 'h': Represents horizontal shifts, determining how far right (positive) or left (negative) the graph moves.
Parameter 'k': Determines vertical shifts, indicating movement up (positive) or down (negative) from the original graph position.