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, 2, right parenthesis(0,2) and left parenthesis, 3, comma, 2000, right parenthesis(3,2000).

Answer 1

To find the exponential function in the form \( y = ab^x \) that passes through the points \( (0, 2) \) and \( (3, 2000) \), we can proceed step by step.

1. **Use the first point (0, 2)**:
From the point \( (0, 2) \), we plug in \( x = 0 \) and \( y = 2 \):
\[
2 = ab^0
\]
Since \( b^0 = 1 \), this reduces to:
\[
2 = a
\]

2. **Use the second point (3, 2000)**:
Now, substitute the known value of \( a \) into the equation using the second point \( (3, 2000) \):
\[
2000 = 2b^3
\]
Dividing both sides by 2 gives:
\[
1000 = b^3
\]
To find \( b \), take the cube root of both sides:
\[
b = \sqrt[3]{1000} = 10
\]

3. **Construct the function**:
Now that we have \( a = 2 \) and \( b = 10 \), we can write the exponential function:
\[
y = 2 \cdot 10^x
\]

Thus, the exponential function that goes through the points \( (0, 2) \) and \( (3, 2000) \) is:
\[
\boxed{y = 2 \cdot 10^x}
\]