To write an exponential function in the form [tex]\( y = a b^x \)[/tex] that goes through the points [tex]\((0, 17)\)[/tex] and [tex]\((2, 272)\)[/tex], we can determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] by solving a system of equations. Here's a step-by-step solution:
1. Start with the general form of the exponential function:
[tex]\[
y = a b^x
\][/tex]
2. Use the first point [tex]\((0, 17)\)[/tex]:
When [tex]\( x = 0 \)[/tex], [tex]\( y = 17 \)[/tex]:
[tex]\[
17 = a b^0
\][/tex]
Since [tex]\( b^0 = 1 \)[/tex], this simplifies to:
[tex]\[
17 = a \cdot 1 \implies a = 17
\][/tex]
3. Substitute [tex]\( a = 17 \)[/tex] into the equation and use the second point [tex]\((2, 272)\)[/tex]:
When [tex]\( x = 2 \)[/tex], [tex]\( y = 272 \)[/tex]:
[tex]\[
272 = 17 b^2
\][/tex]
4. Solve for [tex]\( b \)[/tex]:
[tex]\[
b^2 = \frac{272}{17}
\][/tex]
Calculate the division:
[tex]\[
b^2 = 16
\][/tex]
Finally, take the square root of both sides:
[tex]\[
b = \sqrt{16} \implies b = 4
\][/tex]
5. Form the exponential function:
Now, we have determined that [tex]\( a = 17 \)[/tex] and [tex]\( b = 4 \)[/tex]. Therefore, the exponential function that passes through the points [tex]\((0, 17)\)[/tex] and [tex]\((2, 272)\)[/tex] is:
[tex]\[
y = 17 \cdot 4^x
\][/tex]
Thus, the exponential function is:
[tex]\[
y = 17 \cdot 4^x
\][/tex]
This function satisfies both given points [tex]\((0, 17)\)[/tex] and [tex]\((2, 272)\)[/tex].
