Question

Find the exponential equation for the sequence: (0,4),(1,7.6),(2,14.44)…

y = ?(?)^x

Answer 1

To find the exponential equation of the given sequence (0, 4), (1, 7.6), (2, 14.44), we'll use the general form of an exponential function, which is:

\[
y = a \cdot b^x
\]

where \( a \) is the initial value when \( x = 0 \), and \( b \) is the growth factor.

1. **Find \( a \)**:
From the first point (0, 4), we can see that when \( x = 0 \):

\[
y = a \cdot b^0 = a
\]

Thus, \( a = 4 \).

2. **Use another point to find \( b \)**:
Now we'll use the point (1, 7.6) to find \( b \):

\[
7.6 = 4 \cdot b^1
\]
\[
b = \frac{7.6}{4} = 1.9
\]

3. **Check with the next point (2, 14.44)**:
Now we will check whether this value of \( b \) works with the third point (2, 14.44):

\[
y = 4 \cdot (1.9)^2
\]
\[
y = 4 \cdot 3.61 = 14.44
\]

This matches the third point perfectly.

Thus, the exponential equation for the sequence is:

\[
y = 4 \cdot (1.9)^x
\]