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In this worksheet, you're working with **transformations of exponential functions**, where the base function is \( f(x) = 2^x \), and the transformed function follows the form \( g(x) = a \cdot 2^{k(x - d)} + c \).

Let’s go through the questions and explain the steps to sketch the graphs based on the transformations given:

### Part a) \( h(x) = 3 \cdot 2^{x - 2} + 1 \)

- **Transformation breakdown:**
  - **Vertical stretch**: By a factor of 3 (due to the coefficient 3).
  - **Horizontal shift**: To the right by 2 units (since \( x - 2 \)).
  - **Vertical shift**: Up by 1 unit (due to the \( +1 \)).
  
- **Steps to graph**:
  1. Start with the base function \( 2^x \), then apply the horizontal shift.
  2. Apply the vertical stretch by multiplying the \( y \)-values by 3.
  3. Finally, shift the graph up by 1 unit.

### Part b) \( f(x) = 3 \cdot 2^{x - 1} + 1 \)

- **Transformation breakdown:**
  - Similar to part a, but the horizontal shift is 1 unit to the right.
  
- **Steps to graph**:
  1. Shift the base function \( 2^x \) 1 unit to the right.
  2. Stretch vertically by multiplying the \( y \)-values by 3.
  3. Shift the graph up by 1 unit.

### Part c) \( k(x) = 3 \cdot 2^{-x + 2} + 1 \)

- **Transformation breakdown**:
  - **Horizontal reflection**: Because of the negative sign in front of \( x \), this is a reflection across the y-axis.
  - **Horizontal shift**: Left by 2 units.
  - **Vertical shift**: Up by 1 unit.
  - **Vertical stretch**: Multiply by 3.
  
- **Steps to graph**:
  1. Reflect the base function \( 2^x \) horizontally.
  2. Shift it 2 units left.
  3. Stretch vertically and shift up.

### Part d) \( m(x) = \frac{3}{4} \cdot 2^x + 1 \)

- **Transformation breakdown**:
  - **Vertical compression**: By a factor of \( \frac{3}{4} \) (since the coefficient is less than 1).
  - **Vertical shift**: Up by 1 unit.
  
- **Steps to graph**:
  1. Compress the base function vertically by multiplying the \( y \)-values by \( \frac{3}{4} \).
  2. Shift the graph up by 1 unit.

### Part e) \( n(x) = \frac{1}{2^{x + 2}} + 1 \)

- **Transformation breakdown**:
  - This can be rewritten as \( n(x) = \frac{1}{2^{x}} \cdot \frac{1}{2^2} + 1 \), indicating a horizontal shift left by 2 units and a reflection (because \( \frac{1}{2^x} \) decreases as \( x \) increases).
  
- **Steps to graph**:
  1. Reflect the base function.
  2. Shift it 2 units to the left.
  3. Shift up by 1.

### Part f) \( p(x) = 3 \cdot \frac{1}{2^{-x + 2}} + 1 \)

- **Transformation breakdown**:
  - Reflect across the y-axis and shift right by 2 units (from \( -x + 2 \)).
  - Vertical stretch and shift up.

Would you like detailed step-by-step graphs for each transformation?

### 5 Relative Questions:
1. How does a horizontal shift affect the exponential graph's general behavior?
2. What effect does a vertical stretch or compression have on the base function?
3. How does reflecting an exponential function across the y-axis change its shape?
4. Why does a shift up or down affect the asymptote of an exponential graph?
5. How can we determine the horizontal asymptote from the transformed equation?

### Tip:
Always identify the asymptote after transformations, especially for exponential functions, as it helps you understand the graph’s end behavior.

Transformations of Exponential Functions: Use mapping notation to sketch each graph.

Solve exponential function transformations with steps for applying vertical stretch, horizontal shifts, and reflections. Learn how to sketch each graph for functions like g(x) = 3(2^(x-2)) + 1. Suitable for Grades 10-12.

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