Question

the table represents some points on the graph of an exponential function. which function represents the relationship?
x: 0,1,2,3,4
f(x): 6,3,1.5,0.75,0.375
a y=24(0.5)^x
b y=6(0.5)^x
c y=24(5)^x
d y=0.5^x

Answer 1

To find the function that represents the relationship shown in the table, we can observe the values of \( f(x) \) as \( x \) increases. The values are:

- \( f(0) = 6 \)
- \( f(1) = 3 \)
- \( f(2) = 1.5 \)
- \( f(3) = 0.75 \)
- \( f(4) = 0.375 \)

It appears that as \( x \) increases, \( f(x) \) is decreasing by half at each step. This pattern suggests that the function has a base of \( 0.5 \).

Now we can verify which of the given options fits this pattern:

1. **Option A: \( y = 24(0.5)^x \)**
- \( f(0) = 24(0.5)^0 = 24 \) (not matching)

2. **Option B: \( y = 6(0.5)^x \)**
- \( f(0) = 6(0.5)^0 = 6 \) (matches)
- \( f(1) = 6(0.5)^1 = 3 \) (matches)
- \( f(2) = 6(0.5)^2 = 1.5 \) (matches)
- \( f(3) = 6(0.5)^3 = 0.75 \) (matches)
- \( f(4) = 6(0.5)^4 = 0.375 \) (matches)

3. **Option C: \( y = 24(5)^x \)**
- \( f(0) = 24(5)^0 = 24 \) (not matching)

4. **Option D: \( y = (0.5)^x \)**
- \( f(0) = (0.5)^0 = 1 \) (not matching)

From this analysis, **Option B: \( y = 6(0.5)^x \)** is the function that best represents the relationship in the table.