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.