S of Which function represents a vertical stretch of an exponential function? \( \begin{array}{l}x \quad y=2^{x} \\ y=3 \cdot 2^{x} \\ y=2^{3 x} \\ \text { RETRY }\end{array} \)

A **vertical stretch** of an exponential function occurs when the function is multiplied by a constant factor greater than 1. This scaling affects the \( y \)-values, making the graph "taller" without altering its shape. Let's analyze the given functions: 1. **\( y = 2^x \)** - This is the base exponential function with no stretching or compression. 2. **\( y = 3 \cdot 2^x \)** - Here, the function is multiplied by **3**, which is a constant greater than 1. - **This represents a vertical stretch** of the base exponential function by a factor of 3. 3. **\( y = 2^{3x} \)** - In this case, the exponent is multiplied by **3**, which affects the **horizontal** scaling. - This is a **horizontal compression** by a factor of \( \frac{1}{3} \), not a vertical stretch. 4. **RETRY** - This appears to be an instruction rather than an option and can be disregarded in this context. **Conclusion:** The function that represents a vertical stretch of the exponential function \( y = 2^x \) is: \[ y = 3 \cdot 2^x \] **Graphical Illustration:** - **Base Function (\( y = 2^x \)):** - Passes through (0,1) with a growth rate based on the base 2. - **Vertical Stretch (\( y = 3 \cdot 2^x \)):** - Every \( y \)-value of the base function is multiplied by 3. - Passes through (0,3), making the graph steeper. **Visual Comparison:** | \( x \) | \( y = 2^x \) | \( y = 3 \cdot 2^x \) | |---|---|---| | -1 | 0.5 | 1.5 | | 0 | 1 | 3 | | 1 | 2 | 6 | | 2 | 4 | 12 | | 3 | 8 | 24 | As seen in the table, for each \( x \)-value, the \( y \)-value of \( y = 3 \cdot 2^x \) is three times that of \( y = 2^x \), confirming the vertical stretch. **Final Answer:** \[ y = 3 \cdot 2^x \]