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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?

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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 \(b\).

Write an Exponential Equation that goes through the points (1, 12) and (2, 3). In the equation y = a(b)^x, find the values of a and b, rounding to the nearest hundredth if necessary.

Learn how to determine an exponential equation passing through the points (1, 12) and (2, 3) by solving for constants a and b in the equation y = a(b)^x. Includes step-by-step solution for understanding exponential functions and solving systems of equations. Suitable for high school students in Grades 9-12.

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