Transformation of the Exponential Functions - Algebra-2

1. Fundamental Concepts

Definition: Exponential functions are of the form $$f(x) = a \cdot b^{x}$$ where $$a$$ and $$b$$ are constants, and $$b > 0$$ , $$b \neq 1$$ .
Transformations: Transformations include shifts, stretches, compressions, and reflections of the basic exponential function.
Horizontal Shifts: Adding or subtracting a constant inside the exponent affects horizontal shifts.
Vertical Shifts: Adding or subtracting a constant outside the exponent affects vertical shifts.
Reflections: Multiplying the exponent by -1 reflects the graph over the y-axis; multiplying the entire function by -1 reflects it over the x-axis.
Stretches and Compressions: Multiplying the base by a constant greater than 1 results in a vertical stretch; a constant between 0 and 1 results in a vertical compression.

Basic Rule: $$f(x) = a \cdot b^{x}$$

Horizontal Shift: When, the function shifts h units to the right; when , the function shifts units to the left. $$f(x) = a \cdot b^{x-h}$$

Vertical Shift: When , the function shifts k units upward; when , the function shifts units downward. $$f(x) = a \cdot b^{x} + k$$

Reflection Over Y-Axis: $$f(x) = a \cdot b^{-x}$$

Reflection Over X-Axis: $$f(x) = -a \cdot b^{x}$$

Vertical Stretch/Compression: When, it results in a vertical stretch; when , it leads to a vertical compression.

Problem: Describe the transformations applied to the function $$f(x) = 2^x$$ to obtain $$g(x) = 3 \cdot 2^{x-2} + 4$$ .

The function $$g(x) = 3 \cdot 2^{x-2} + 4$$ involves a horizontal shift right by 2 units, a vertical stretch by a factor of 3, and a vertical shift up by 4 units.

Problem: Write the equation for a function that is a reflection of $$f(x) = 4 \cdot 3^x$$ over the y-axis and then shifted down by 3 units.

Visual Strategy: Graph each transformation step-by-step to visualize changes.
Error-Proofing: Double-check each transformation by substituting values into the original and transformed functions.
Concept Reinforcement: Practice with a variety of functions to understand the effects of different transformations.