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The given function is:

\[
f(x) = 2 \cdot \left( \frac{1}{4} \right)^{x-1} + 3
\]

### Step 1: Identify the Parent Function
The parent function for this equation is an exponential decay function of the form:

\[
f(x) = a^x
\]

In this case, the base is \( a = \frac{1}{4} \), meaning the parent function is:

\[
f(x) = \left( \frac{1}{4} \right)^x
\]

### Step 2: Identify the Transformations

The given function involves several transformations applied to the parent function. Let's break them down:

1. **Horizontal Shift**: The term \( x - 1 \) indicates a shift to the right by 1 unit.
   
2. **Vertical Stretch**: The coefficient \( 2 \) outside the exponential function means the graph is stretched vertically by a factor of 2.

3. **Vertical Shift**: The constant \( +3 \) outside the function indicates a vertical shift upward by 3 units.

Thus, the transformations are:
- Horizontal shift right by 1 unit.
- Vertical stretch by a factor of 2.
- Vertical shift up by 3 units.

### Final Form of the Function:
\[
f(x) = 2 \cdot \left( \frac{1}{4} \right)^{x-1} + 3
\]

Would you like further details or have any questions?

Here are some relative questions for you to consider:
1. How would the function change if the base was \( \frac{1}{2} \) instead of \( \frac{1}{4} \)?
2. What happens if the horizontal shift is replaced with \( x + 1 \)?
3. How do vertical stretches affect the steepness of an exponential graph?
4. What if the vertical shift was \( -3 \) instead of \( +3 \)?
5. How can you determine whether the function represents exponential growth or decay?

**Tip:** Remember that exponential functions with bases between 0 and 1 represent decay, while those with bases greater than 1 represent growth.

Identify the parent function and its transformations: f(x) = 2 · (1/4)^(x-1) + 3.

Learn how to identify the parent function and transformations of the exponential function f(x) = 2 · (1/4)^(x-1) + 3. Explore concepts like exponential decay, horizontal and vertical shifts, and vertical stretching.

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