Which of the following exponential functions passes through the points left parenthesis 0 comma 18 right parenthesis and left parenthesis 12 comma 8 right parenthesis?

		
f left parenthesis x right parenthesis equals 12 open parentheses 2 close parentheses to the power of x divided by 3 end exponent

		
f left parenthesis x right parenthesis equals 18 open parentheses 2 over 3 close parentheses to the power of x divided by 6 end exponent

		
f left parenthesis x right parenthesis equals 12 open parentheses 1 half close parentheses to the power of x divided by 3 end exponent

		
f left parenthesis x right parenthesis equals 18 open parentheses 5 over 4 close parentheses to the power of x divided by 2 end exponent

		
f left parenthesis x right parenthesis equals 12 open parentheses 1 third close parentheses to the power of x divided by 2 end exponent

		
f left parenthesis x right parenthesis equals 18 open parentheses 3 close parentheses to the power of x divided by 5 end exponent



We are tasked with determining which exponential function passes through the points \((0, 18)\) and \((12, 8)\). Let's evaluate each function by checking if it satisfies these two points.

### Steps:
1. **Point (0, 18):**
   - When \(x = 0\), \(f(0)\) should equal 18.
   - Plug \(x = 0\) into each function and check if the result is 18.
  
2. **Point (12, 8):**
   - When \(x = 12\), \(f(12)\) should equal 8.
   - Plug \(x = 12\) into each function and check if the result is 8.

### Evaluate Each Function:

#### 1. \(f(x) = 12 \cdot (2)^{\frac{x}{3}}\)
   - At \(x = 0\): \(f(0) = 12 \cdot (2)^0 = 12\) (does not satisfy \(f(0) = 18\)).
   - No need to check further, this is not the correct function.

#### 2. \(f(x) = 18 \cdot \left(\frac{2}{3}\right)^{\frac{x}{6}}\)
   - At \(x = 0\): \(f(0) = 18 \cdot \left(\frac{2}{3}\right)^0 = 18\) (satisfies \(f(0) = 18\)).
   - At \(x = 12\): 
     \[
     f(12) = 18 \cdot \left(\frac{2}{3}\right)^{\frac{12}{6}} = 18 \cdot \left(\frac{2}{3}\right)^2 = 18 \cdot \frac{4}{9} = 8.
     \]
     (Satisfies \(f(12) = 8\)).
   - This function passes through both points.

#### 3. \(f(x) = 12 \cdot \left(\frac{1}{2}\right)^{\frac{x}{3}}\)
   - At \(x = 0\): \(f(0) = 12 \cdot \left(\frac{1}{2}\right)^0 = 12\) (does not satisfy \(f(0) = 18\)).
   - This is not the correct function.

#### 4. \(f(x) = 18 \cdot \left(\frac{5}{4}\right)^{\frac{x}{2}}\)
   - At \(x = 0\): \(f(0) = 18 \cdot \left(\frac{5}{4}\right)^0 = 18\) (satisfies \(f(0) = 18\)).
   - At \(x = 12\): 
     \[
     f(12) = 18 \cdot \left(\frac{5}{4}\right)^6 \approx 18 \cdot 7.669 \approx 138.
     \]
     (Does not satisfy \(f(12) = 8\)).

#### 5. \(f(x) = 12 \cdot \left(\frac{1}{3}\right)^{\frac{x}{2}}\)
   - At \(x = 0\): \(f(0) = 12 \cdot \left(\frac{1}{3}\right)^0 = 12\) (does not satisfy \(f(0) = 18\)).
   - This is not the correct function.

#### 6. \(f(x) = 18 \cdot (3)^{\frac{x}{5}}\)
   - At \(x = 0\): \(f(0) = 18 \cdot (3)^0 = 18\) (satisfies \(f(0) = 18\)).
   - At \(x = 12\): 
     \[
     f(12) = 18 \cdot (3)^{\frac{12}{5}} \approx 18 \cdot 10.627 \approx 191.
     \]
     (Does not satisfy \(f(12) = 8\)).

### Conclusion:
The correct function is:
\[
f(x) = 18 \cdot \left(\frac{2}{3}\right)^{\frac{x}{6}}.
\]

Would you like further explanation or have any questions? Here are some related questions:

1. How do exponential functions change with different bases?
2. What role do the exponents play in controlling the growth/decay rate?
3. How can you solve exponential equations involving logarithms?
4. What are the applications of exponential functions in real-world scenarios?
5. How would you graph this exponential function?

**Tip:** Exponential functions with a base between 0 and 1 model decay, while those with a base greater than 1 model growth.

Which of the following exponential functions passes through the points (0, 18) and (12, 8)?

Discover which exponential function passes through the points (0, 18) and (12, 8) by evaluating different options. This problem involves key concepts like function evaluation, exponential growth/decay, and interpreting coordinates.

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