Question

The table contains some points on the graph of an exponential function.


x
y
0 0.125
1 0.5
2 2
3 8



Based on the table, which function represents the same relationship?
Responses
A p(x) = (0.25)xp ( x ) = (0.25) x
B p(x) = 32(0.25)xp ( x ) = 32(0.25) x
C p(x) = 0.5(8)xp ( x ) = 0.5(8) x
D p(x) = 0.125(4)x

Answer 1

To identify the exponential function that represents the relationship in the given table, we can analyze the points and look for a consistent pattern.

The points provided are:

- \( (0, 0.125) \)
- \( (1, 0.5) \)
- \( (2, 2) \)
- \( (3, 8) \)

We can express the exponential function in the general form \( p(x) = a \cdot b^x \), where \( a \) is the initial value at \( x = 0 \) and \( b \) is the base of the exponential.

From the point \( (0, 0.125) \), we see that when \( x = 0 \), \( p(0) = a = 0.125 \).

To find \( b \), we can analyze one of the other points. Let's use the point \( (1, 0.5) \):

\[
p(1) = 0.125 \cdot b = 0.5
\]

Solving for \( b \):

\[
b = \frac{0.5}{0.125} = 4
\]

Now, let's check this with another point, such as \( (2, 2) \):

\[
p(2) = 0.125 \cdot 4^2 = 0.125 \cdot 16 = 2
\]

This is consistent. Now, let's check the point \( (3, 8) \):

\[
p(3) = 0.125 \cdot 4^3 = 0.125 \cdot 64 = 8
\]

This also checks out. Therefore, the exponential function is:

\[
p(x) = 0.125 \cdot 4^x
\]

Now, looking at the options provided:

A) \( p(x) = (0.25)^x \)

B) \( p(x) = 32(0.25)^x \)

C) \( p(x) = 0.5(8)^x \)

D) \( p(x) = 0.125(4)^x \)

The function we derived matches option D:

**D) \( p(x) = 0.125(4)^x \)**

Thus, the correct response is D.