Question

The table contains some points on the graph of an exponential function.
X
0
1
2
3
0.0625
0.25
1
4
Based on the table, which function represents the same relationship?
A. q(x) = (0.25)*
B. q(x) = 256(0.25)*
C. q(x)=0.0625(4)*
D. q(x)=0.5(4)*

Answer 1

To identify the function representing the relationship in the given points of the exponential function, we can analyze the points provided, which seem to correspond to values of \( f(x) = a \cdot b^x \).

The given points are:
- For \( x = 0 \), \( y = 0.0625 \)
- For \( x = 1 \), \( y = 0.25 \)
- For \( x = 2 \), \( y = 1 \)
- For \( x = 3 \), \( y = 4 \)

Assuming the function has the form \( f(x) = a \cdot b^x \), the point \( (0, 0.0625) \) gives us:

\[
f(0) = a \cdot b^0 = a = 0.0625
\]

Using the next point \( (1, 0.25) \):

\[
f(1) = 0.0625 \cdot b = 0.25
\]

Solving for \( b \):

\[
b = \frac{0.25}{0.0625} = 4
\]

Now we have:

\[
f(x) = 0.0625 \cdot 4^x
\]

Let's check if this matches the next points:

For \( x = 2 \):

\[
f(2) = 0.0625 \cdot 4^2 = 0.0625 \cdot 16 = 1
\]

For \( x = 3 \):

\[
f(3) = 0.0625 \cdot 4^3 = 0.0625 \cdot 64 = 4
\]

Now, let's look for an option that matches this function:

A. \( q(x) = 0.25 \cdot b^x \)
B. \( q(x) = 256(0.25)^x \)
C. \( q(x) = 0.0625(4)^x \)
D. \( q(x) = 0.5(4)^x \)

The function we've derived, \( q(x) = 0.0625(4)^x \), corresponds to option C.

Thus, the correct answer is:

**C. \( q(x) = 0.0625(4)^x \)**