Transformations of Logarithmic Functions - Algebra-2

1. Fundamental Concepts

Definition: Transformations of logarithmic functions involve modifying the basic form $$y = \log_b(x)$$ through shifts, stretches, and reflections.
Shifts: Horizontal and vertical shifts change the position of the graph without altering its shape.
Stretches/Compressions: These transformations affect the steepness of the graph by multiplying the function or the input by a constant.
Reflections: Reflections over the x-axis or y-axis invert the graph across these axes.

2. Key Concepts

Horizontal Shift: $$y = \log_b(x - h)$$

A shift to the right if $$h > 0$$ and to the left if $$h < 0$$.

Vertical Shift: $$y = \log_b(x) + k$$

A shift upward if $$k > 0$$ and downward if $$k < 0$$.

Vertical Stretch/Compression: $$y = a \cdot \log_b(x)$$

The graph is stretched vertically if $$|a| > 1$$ and compressed if $$0 < |a| < 1$$.

Reflection Over the x-Axis: $$y = -\log_b(x)$$

The graph is reflected over the x-axis.

3. Examples

Example 1 (Basic)

Problem: Describe the transformation of the function $$y = \log_2(x)$$ to $$y = \log_2(x - 3) + 2$$.

Step-by-Step Solution:

The term $$x - 3$$ indicates a horizontal shift to the right by 3 units.
The term $$+ 2$$ indicates a vertical shift upward by 2 units.

Validation: The original function $$y = \log_2(x)$$ has its vertical asymptote at $$x = 0$$. After the transformation, the new function $$y = \log_2(x - 3) + 2$$ has its vertical asymptote at $$x = 3$$ and is shifted up by 2 units.

Example 2 (Intermediate)

Problem: Graph the function $$y = -2 \cdot \log_3(x + 1) - 1$$ and describe the transformations applied to the parent function $$y = \log_3(x)$$.

Step-by-Step Solution:

The term $$x + 1$$ indicates a horizontal shift to the left by 1 unit.
The term $$-2$$ indicates a vertical stretch by a factor of 2 and a reflection over the x-axis.
The term $$- 1$$ indicates a vertical shift downward by 1 unit.

Validation: The original function $$y = \log_3(x)$$ has its vertical asymptote at $$x = 0$$. After the transformations, the new function $$y = -2 \cdot \log_3(x + 1) - 1$$ has its vertical asymptote at $$x = -1$$, is reflected over the x-axis, stretched vertically by a factor of 2, and shifted down by 1 unit.

4. Problem-Solving Techniques

Graphical Analysis: Use graphs to visualize the effects of each transformation on the parent function.
Step-by-Step Approach: Break down the problem into individual transformations and apply them sequentially.
Substitution Method: Substitute specific values for $$x$$ to determine the impact of transformations on key points of the graph.