Practice Examples: Transformations of Exponential Functions

📚 Topic Summary

Transformations of exponential functions involve altering the basic exponential function, $f(x) = a^x$, through shifts, stretches, compressions, and reflections. These transformations change the graph's position, shape, and orientation. Understanding these transformations allows us to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. By recognizing the impact of each transformation parameter, we can easily analyze and predict the behavior of exponential functions in different contexts.

The general form of a transformed exponential function is $g(x) = A \cdot a^{B(x - C)} + D$, where:

• 🔍 $A$ represents a vertical stretch or compression and reflection across the x-axis if $A < 0$.
• 📈 $B$ represents a horizontal stretch or compression and reflection across the y-axis if $B < 0$.
• ➡️ $C$ represents a horizontal shift.
• ⬆️ $D$ represents a vertical shift.

🧮 Part A: Vocabulary

Match the term with its correct definition:

1. Vertical Stretch
2. Horizontal Shift
3. Reflection across x-axis
4. Exponential Function
5. Vertical Compression

1. A function of the form $f(x) = a^x$, where $a$ is a constant.
2. A transformation that moves the graph left or right.
3. A transformation that shrinks the graph vertically.
4. A transformation that flips the graph over the x-axis.
5. A transformation that stretches the graph vertically.

Answers:

• Vertical Stretch - A transformation that stretches the graph vertically.
• Horizontal Shift - A transformation that moves the graph left or right.
• Reflection across x-axis - A transformation that flips the graph over the x-axis.
• Exponential Function - A function of the form $f(x) = a^x$, where $a$ is a constant.
• Vertical Compression - A transformation that shrinks the graph vertically.

✍️ Part B: Fill in the Blanks

Complete the following paragraph with the correct terms:

The general form of a transformed exponential function is $g(x) = A \cdot a^{B(x - C)} + D$. The parameter _______ represents a vertical stretch or compression, while _______ represents a horizontal shift. If $A$ is negative, there is a _______ across the x-axis. The parameter $D$ represents a _______ shift.

Answers:

• A
• C
• Reflection
• Vertical

🤔 Part C: Critical Thinking

Explain how changing the base 'a' in the exponential function $f(x) = a^x$ affects the graph's behavior. Provide examples to support your explanation.