To determine which of the given functions fits the exponential model passing through the points $(0, 3)$ and $(2, 12)$, let's analyze step by step:

Step 1: Identify the general form of exponential functions.

Exponential functions are generally written as: $y = a \cdot b^x$ where:

• $a$ is the initial value (when $x = 0$),
• $b$ is the base (growth or decay factor).

Step 2: Use the point $(0, 3)$.

Substitute $x = 0$ and $y = 3$ into $y = a \cdot b^x$: $3 = a \cdot b^0$ Since $b^0 = 1$, this simplifies to: $a = 3$ Thus, the equation becomes: $y = 3 \cdot b^x$

Step 3: Use the point $(2, 12)$.

Substitute $x = 2$ and $y = 12$ into $y = 3 \cdot b^x$: $12 = 3 \cdot b^2$ Divide both sides by 3: $4 = b^2$ Solve for $b$: $b = 2$ Thus, the equation becomes: $y = 3 \cdot 2^x$

Step 4: Match with the options.

From the given options, the correct choice is: $\boxed{\text{(B) } y = 3 \cdot 2^x}$

Would you like further details on any step or related concepts?

Here are 5 related questions to enhance understanding:

1. How do you derive the equation of an exponential function from two given points?
2. What distinguishes exponential growth functions from linear or sinusoidal functions?
3. How can you verify whether a point lies on a given exponential curve?
4. What are the characteristics of the base $b$ in exponential functions, and what happens when $b > 1$ or $0 < b < 1$?
5. How do exponential functions apply in real-world scenarios, such as population growth or radioactive decay?

Tip: To verify solutions involving functions, always substitute the given points into the derived equation to ensure accuracy!