To find an exponential equation that passes through the points $(1, 12)$ and $(2, 3)$, we’ll use the form of the exponential function:

$y = a \cdot b^x$

Step 1: Substitute the Points into the Equation

For each point, substitute $x$ and $y$ into the equation $y = a \cdot b^x$ to get two equations.

1. For the point $(1, 12)$: $12 = a \cdot b^1 \Rightarrow 12 = a \cdot b$

2. For the point $(2, 3)$: $3 = a \cdot b^2$

Step 2: Set Up a System of Equations

We now have the following system of equations: $12 = a \cdot b$ $3 = a \cdot b^2$

Step 3: Solve for $b$

Divide the second equation by the first equation to eliminate $a$: $\frac{3}{12} = \frac{a \cdot b^2}{a \cdot b}$ This simplifies to: $\frac{1}{4} = b$

So, $b = 0.25$.

Step 4: Solve for $a$

Substitute $b = 0.25$ into the first equation: $12 = a \cdot 0.25$ $a = \frac{12}{0.25} = 48$

Final Answer

The values are: $a = 48 \quad \text{and} \quad b = 0.25$

Thus, the exponential equation is: $y = 48 \cdot (0.25)^x$

Would you like further details on each step?

Here are five additional questions for practice:

1. What is the general form of an exponential function?
2. How would you find an exponential function given two points with negative $y$-values?
3. What happens to the exponential function if $b$ is greater than 1?
4. How do you interpret the values of $a$ and $b$ in real-world applications of exponential functions?
5. How do you solve exponential functions when given three points?

Tip: For exponential functions, if the ratio of outputs at successive points is constant, it helps to directly find the base $b$.