To identify which of the functions represent exponential growth, we need to consider the general form of an exponential function:

$y = a(b)^x$ Where:

• $a$ is the initial value (can be positive or negative),
• $b$ is the base, and
• $x$ is the exponent.

For exponential growth, the base $b$ must be greater than 1. If $b > 1$, the function grows as $x$ increases. Conversely, if $0 < b < 1$, the function represents exponential decay.

Let's analyze each option:

1. $y = 8(4)^x$:

   • Here, $b = 4$ (which is greater than 1), so this represents exponential growth.

2. $y = 2 \left(\frac{5}{7}\right)^x$:

   • Here, $b = \frac{5}{7}$, and since $\frac{5}{7} < 1$, this represents exponential decay, not growth.

3. $y = 0.2(1.4)^x$:

   • Here, $b = 1.4$ (which is greater than 1), so this represents exponential growth.

4. $y = \frac{3}{4}(0.99)^x$:

   • Here, $b = 0.99$, and since $0.99 < 1$, this represents exponential decay, not growth.

5. $y = \frac{1}{2}(1.01)^x$:

   • Here, $b = 1.01$ (which is slightly greater than 1), so this represents exponential growth.

Final Answer:

The functions that represent exponential growth are:

• $y = 8(4)^x$
• $y = 0.2(1.4)^x$
• $y = \frac{1}{2}(1.01)^x$

Would you like more details on how exponential growth works, or do you have any questions?

Here are five related questions to further explore this topic:

1. What is the difference between exponential growth and exponential decay?
2. How does changing the base $b$ in the exponential function affect the growth rate?
3. Can exponential functions model real-world scenarios? If so, can you give examples?
4. How do you graph an exponential growth function, and what are its key features?
5. What happens to the function if the initial value $a$ is negative in an exponential function?

Tip: Always check the base $b$ when analyzing exponential functions. If it's greater than 1, it's exponential growth; if between 0 and 1, it's exponential decay.