To determine which function represents a vertical stretch of an exponential function, let's review the concept of vertical stretching in exponential functions. That occurs when the function is multiplied by a constant factor (other than 1) that scales the function vertically.
Let's analyze each of the given functions:
1. [tex]\( f(x) = 3\left(\frac{1}{2}\right)^x \)[/tex]
- In this function, we have a constant multiplier [tex]\(3\)[/tex] in front of the exponential term [tex]\(\left(\frac{1}{2}\right)^x\)[/tex]. This multiplier scales the output of the exponential function vertically. Therefore, this represents a vertical stretch.
2. [tex]\( f(x) = \frac{1}{2}(3)^x \)[/tex]
- Here, the exponential function is [tex]\( 3^x \)[/tex] and is multiplied by [tex]\(\frac{1}{2}\)[/tex]. This constant [tex]\(\frac{1}{2}\)[/tex] scales the function vertically down (which is also a type of vertical scaling). However, it's not traditionally referred to as a stretch but rather a compression or a shrink.
3. [tex]\( f(x) = (3)^{2x} \)[/tex]
- This function involves modifying the exponent itself by multiplying [tex]\(x\)[/tex] by [tex]\(2\)[/tex]. It does not fit the description of vertical stretching as it’s actually affecting the rate of growth or decay horizontally.
4. [tex]\( f(x) = 3^{\left(\frac{1}{2} x\right)} \)[/tex]
- Here, the exponent is being scaled, making the function grow slower horizontally. This is an example of a horizontal scaling (specifically, a horizontal stretch).
Considering these forms, the function that clearly represents a vertical stretch is:
[tex]\[ f(x) = 3\left(\frac{1}{2}\right)^x \][/tex]
Thus, the correct function that represents a vertical stretch of an exponential function is:
[tex]\[ f(x) = 3\left(\frac{1}{2}\right)^x \][/tex]
