Question

Write an exponential function in the form y, equals, a, b, to the power xy=ab
x
that goes through the points left parenthesis, 0, comma, 8, right parenthesis(0,8) and left parenthesis, 2, comma, 200, right parenthesis(2,200).

Answer 1

An exponential function can be represented in the form:

\[
y = ab^x
\]

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

Given the points \( (0, 8) \) and \( (2, 200) \):

1. From the point \( (0, 8) \), we substitute \( x = 0 \) into the function:

\[
y = ab^0 = a
\]

Thus, we have \( a = 8 \).

2. Now, substitute \( a \) into the function and use the second point \( (2, 200) \):

\[
y = 8b^x
\]

Using the point \( (2, 200) \):

\[
200 = 8b^2
\]

Now solve for \( b^2 \):

\[
b^2 = \frac{200}{8} = 25
\]

Taking the square root of both sides, we find:

\[
b = 5
\]

Now we can write the exponential function:

\[
y = 8 \cdot 5^x
\]

Thus, the exponential function that goes through the points \( (0, 8) \) and \( (2, 200) \) is:

\[
\boxed{y = 8 \cdot 5^x}
\]