Question

The table represents some points on the graph of an exponential function. Which function represents the relationship?
x = 2, 3, 4, 5, 6
f(x) = 6, 3, 1.5, 0.75, 0.375
a. y=24\left(0.5\right)^{x}
b. y=6\left(0.5\right)^{x}
c. y=24\left(5\right)^{x}
d. y=0.5^{x}

Answer 1

To determine which function represents the relationship shown in the table, we can analyze the values given:

- For \( x = 2 \), \( f(x) = 6 \)
- For \( x = 3 \), \( f(x) = 3 \)
- For \( x = 4 \), \( f(x) = 1.5 \)
- For \( x = 5 \), \( f(x) = 0.75 \)
- For \( x = 6 \), \( f(x) = 0.375 \)

Looking closely, we can see that as \( x \) increases, \( f(x) \) decreases. This indicates we are likely dealing with a decreasing exponential function.

### Finding the Exponential Function

Let's assume a general form of the function is:

\[ f(x) = a \cdot b^x \]

From the values provided, we can see that when \( x = 2 \), \( f(2) = 6 \). We can start testing each option to determine which fits.

### Testing Each Choice

1. **Option a: \( y = 24(0.5)^x \)**
- \( f(2) = 24(0.5)^2 = 24 \cdot 0.25 = 6 \) ✓ (matches)
- \( f(3) = 24(0.5)^3 = 24 \cdot 0.125 = 3 \) ✓ (matches)
- \( f(4) = 24(0.5)^4 = 24 \cdot 0.0625 = 1.5 \) ✓ (matches)
- \( f(5) = 24(0.5)^5 = 24 \cdot 0.03125 = 0.75 \) ✓ (matches)
- \( f(6) = 24(0.5)^6 = 24 \cdot 0.015625 = 0.375 \) ✓ (matches)

Since all values match, this function appears to represent the relationship.

2. **Other Options:**

Just for thoroughness, let's check the effectiveness of the other functions:

- **Option b: \( y = 6(0.5)^x \)**
- This won't match for \( x=2 \), since \( f(2) = 6(0.5)^2 = 1.5 \).
- **Option c: \( y = 24(5)^x \)**
- This will produce rapidly increasing values rather than decreasing ones.
- **Option d: \( y = 0.5^x \)**
- Would provide values starting from 1 and not matching any \( f(x) \) given.

### Conclusion:

The function that matches all provided values in the table is:

**a. \( y = 24(0.5)^x \)**.