📚 Topic Summary
Graphing logarithmic function transformations involves understanding how changes to the basic logarithmic function, $f(x) = \log_b(x)$, affect its graph. These transformations include vertical and horizontal shifts, stretches, compressions, and reflections. By recognizing how each parameter in the transformed function, such as $f(x) = a\log_b(x-h) + k$, influences the graph, you can accurately sketch and analyze logarithmic functions. Remember that $a$ controls vertical stretch/compression and reflection, $h$ controls horizontal shift, and $k$ controls vertical shift. Understanding the parent function and the effect of each transformation will make graphing logarithmic function transformations easier.
🧠 Part A: Vocabulary
Match the term to its correct definition:
| Term |
Definition |
| 1. Asymptote |
A. A transformation that flips a graph over a line. |
| 2. Reflection |
B. The inverse of an exponential function. |
| 3. Vertical Shift |
C. A line that a curve approaches but does not intersect. |
| 4. Logarithmic Function |
D. A transformation that moves a graph up or down. |
| 5. Horizontal Shift |
E. A transformation that moves a graph left or right. |
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms.
The general form of a transformed logarithmic function is $f(x) = a\log_b(x - h) + k$. The parameter 'a' controls the vertical ______ and/or reflection. The parameter 'h' represents a horizontal ______, and the parameter 'k' represents a vertical ______. If 'a' is negative, the graph is reflected over the ______-axis.
🤔 Part C: Critical Thinking
Describe how the graph of $f(x) = -2\log_3(x + 1) - 4$ is transformed from the parent function $g(x) = \log_3(x)$. Include specific details about each transformation.
