Algebra 2 Unit 1 Review: Transformations

Transformations in Algebra 2

• Horizontal Shrink: Occurs when x-values are multiplied by a number greater than 1 inside the function f(ax).
• Vertical Shrink: Achieved by multiplying the function by a fraction (less than 1) on the outside, affecting y-values as af(x).
• Vertical Stretch: Involves multiplying the function by a number greater than 1 on the outside, which also affects y-values as af(x).
• Horizontal Stretch: Happens when x-values are multiplied by a fraction (less than 1) inside the function f(ax).

Parent Function Domain & Range

• Domain & Range [0, ∞): Functions that have a domain starting from 0 and extending to infinity, reflecting graph characteristics like the Square Root or Absolute Value functions.
• Domain & Range (-∞, ∞): Functions with unrestricted x-values and y-values, such as the Linear, Cubic, and Cube Root parent functions.
• Absolute Value Function: Has a range of [0, ∞) with the equation f(x) = |x|.

Movement of Functions

• Movement Up: Achieved by adding a constant outside the function (f(x) + k).
• Movement Down: Involves subtracting a constant outside the function (f(x) - k).
• Move Left: Addition inside the function (f(x + k)) shifts the graph to the left.
• Move Right: Subtracting a constant within the function (f(x - k)) shifts the graph to the right.

Parent Functions

• Linear Parent Function: Represented by f(x) = x; domain is (-∞, ∞) and range is (-∞, ∞).
• Quadratic Parent Function: Typically represented by f(x) = x²; domain is (-∞, ∞) and range is [0, ∞).
• Cubic Parent Function: Defined by f(x) = x³; domain and range are both (-∞, ∞).
• Cube Root Parent Function: Denoted by f(x) = ∛x; domain and range are (-∞, ∞).
• Square Root Parent Function: Given by f(x) = √x; domain is [0, ∞) and range is also [0, ∞).
• Absolute Value Parent Function: Expressed as f(x) = |x|; domain is (-∞, ∞) and range is [0, ∞).
• Exponential Parent Function: Defined as f(x) = a^x, generally with domain (-∞, ∞) and range (0, ∞).

Reflections

• Reflects Over x-axis: Achieved by multiplying the function by -1 outside (-f(x)).
• Reflect Over y-axis: Implemented by multiplying the x-values by -1 inside the function (f(-x)).

Inverse Functions

• Defined by the condition that if f(g(x)) = g(f(x)) = x, then f(x) and g(x) are considered inverses of each other.

General Concept of Domain

• Domain refers to all possible x-values that a function can accept.