To determine which function represents a vertical stretch of an exponential function, let's consider the original form of an exponential function, which can be written as \( f(x) = a \cdot b^x \), where \( a \) and \( b \) are constants. A vertical stretch involves multiplying the entire function by a constant factor greater than 1.
Let's examine each option provided:
1. Option 1: \( f(x) = 3 \left( \frac{1}{2} \right)^x \)
- Here, the base of the exponential function is \( \frac{1}{2} \), and the function is multiplied by 3.
- The factor \( 3 \) serves as the vertical stretch of the original exponential function.
2. Option 2: \( f(x) = \frac{1}{2} (3)^x \)
- In this case, the base of the exponential function is \( 3 \), and it is multiplied by \( \frac{1}{2} \).
- Multiplying by \( \frac{1}{2} \) actually compresses the function vertically, not stretches it.
3. Option 3: \( f(x) = (3)^{2x} \)
- This function suggests an exponent \( 2x \), which affects the rate of growth of the function but does not represent a vertical stretch.
- Changing the exponent in this way accelerates the growth or decay of the function, rather than multiplying the whole function to stretch it.
4. Option 4: \( f(x) = 3^{\left(\frac{1}{2} x\right)} \)
- This expression involves modifying the exponent to \( \frac{1}{2} x \), which slows down the growth rate of the function.
- Again, this does not represent a vertical stretch of the function but changes the rate at which it grows.
From this analysis, it is clear that the function in Option 1: \( f(x) = 3 \left( \frac{1}{2} \right)^x \) represents a vertical stretch of an exponential function by a factor of 3. Therefore, the correct answer is:
1
