The function given is $y = 3 \cdot 4^x$. We are asked to evaluate this function for $x = -2$, $x = 0$, and $x = 3$, and to show the work for each calculation. Let's go through each value of $x$ step-by-step:

For $x = -2$:

$y = 3 \cdot 4^{-2}$ Using the property of exponents, $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$: $y = 3 \cdot \frac{1}{16} = \frac{3}{16}$

For $x = 0$:

$y = 3 \cdot 4^0$ Since any number raised to the power of 0 is 1: $y = 3 \cdot 1 = 3$

For $x = 3$:

$y = 3 \cdot 4^3$ Calculate $4^3 = 64$: $y = 3 \cdot 64 = 192$

Summary of Results:

• For $x = -2$: $y = \frac{3}{16}$
• For $x = 0$: $y = 3$
• For $x = 3$: $y = 192$

Would you like a more detailed explanation of any part of this process, or do you have additional questions?

Here are 5 related questions to further explore this concept:

1. How does changing the base in an exponential function affect the results?
2. What happens if you change the coefficient in front of the exponential function?
3. How would the function behave for negative values of the base?
4. Can you explain why any number raised to the power of 0 is always 1?
5. What other applications of exponential functions can you find in real-world scenarios?

Tip: When evaluating exponential functions, remember to handle negative exponents by converting them to fractions for easier calculations.