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, 10, right parenthesis(0,10) and left parenthesis, 2, comma, 160, right parenthesis(2,160).

Answer 1

To find an exponential function in the form \( y = a \cdot b^x \) that goes through the points \( (0, 10) \) and \( (2, 160) \), we can use these points to establish a system of equations.

1. Plugging in the first point \( (0, 10) \):

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

Using \( x = 0 \) and \( y = 10 \):

\[
10 = a \cdot b^0 \implies 10 = a \cdot 1 \implies a = 10
\]

2. With \( a = 10 \), we can now use the second point \( (2, 160) \):

\[
160 = 10 \cdot b^2
\]

To solve for \( b^2 \):

\[
b^2 = \frac{160}{10} = 16
\]

Taking the square root of both sides:

\[
b = \sqrt{16} = 4
\]

Now we have \( a \) and \( b \):

- \( a = 10 \)
- \( b = 4 \)

Thus, the exponential function is:

\[
y = 10 \cdot 4^x
\]

### Final Function:

\[
\boxed{y = 10 \cdot 4^x}
\]