Question

The table represents some points on the graph of an exponential function. Which function represents the relationship?

The table
x: 2 ,3 ,4 ,5, 6
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 identify the correct exponential function from the options given, we can start by examining the values provided in the table.

The table contains the following pairs of values:

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

From the observations:

- When \( x \) increases by 1, \( f(x) \) appears to be halved:
- \( f(3) = 3 \) is half of \( f(2) = 6 \)
- \( f(4) = 1.5 \) is half of \( f(3) = 3 \)
- \( f(5) = 0.75 \) is half of \( f(4) = 1.5 \)
- \( f(6) = 0.375 \) is half of \( f(5) = 0.75 \)

This pattern suggests that the function can be represented in the form \( y = a(0.5)^x \).

Next, we need to determine the value of \( a \). We know that at \( x = 2 \), \( f(2) = 6 \).

Using the equation:

\[
f(2) = a(0.5)^2
\]

Substituting the value:

\[
6 = a(0.25)
\]

Solving for \( a \):

\[
a = \frac{6}{0.25} = 6 \times 4 = 24
\]

Thus, the function is:

\[
f(x) = 24(0.5)^x
\]

Now, let's match this with the options provided:

A: \( y = 24(0.5)^x \)
B: \( y = 6(0.5)^x \)
C: \( y = 24(5)^x \)
D: \( y = 0.5^x \)

The function that matches is **A: \( y = 24(0.5)^x \)**.