Question

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

x y
0 0.0625
1 0.25
2 1
3 4

Based on the table, which function represents the same relationship?
A. ( ) (0.25)x q x =
B. ( ) 256(0.25)x q x =
C. ( ) 0.0625(4)x q x =
D. ( ) 0.5(4)x q x =

Answer 1

To determine which function represents the exponential relationship based on the given points in the table, let's analyze the data and compare it against the provided options.

The points from the table indicate the following (in the form of \((x, y)\)):
- (0, 0.0625)
- (1, 0.25)
- (2, 1)
- (3, 4)

We can observe how the \(y\)-values change with respect to \(x\):
- When \(x = 0\), \(y = 0.0625 = \frac{1}{16}\)
- When \(x = 1\), \(y = 0.25 = \frac{1}{4}\)
- When \(x = 2\), \(y = 1 = \frac{1}{1}\)
- When \(x = 3\), \(y = 4 = \frac{4}{1}\)

From the sequence of \(y\)-values, it appears that:
- \(y\) is increasing as \(x\) increases.

Using the values, we can derive an exponential function in the form \(y = ab^x\) where \(a\) is the initial value at \(x=0\) and \(b\) is the growth factor.

We find \(y\) when \(x=0\) is \(0.0625\):
So, \(a = 0.0625\).

To find \(b\), we can look at the ratio of consecutive \(y\) values:
- From \(x=0\) to \(x=1\): \( \frac{0.25}{0.0625} = 4\)
- From \(x=1\) to \(x=2\): \( \frac{1}{0.25} = 4\)
- From \(x=2\) to \(x=3\): \( \frac{4}{1} = 4\)

Since all of these ratios equal \(4\), the growth factor \(b = 4\).

Thus, we have derived that the function in the form:

\[
y = 0.0625 \cdot 4^x
\]

This corresponds to the function option C:
\[
( ) 0.0625(4)^x = q(x)
\]

Therefore, the answer is:
**C. ( ) 0.0625(4)^x = q(x)**