Question

Find the formula for an exponential function that passes through the two points given. (x, y) = (-1, 9/2) and (x, y) = (3, 72)

f(x)=​

Answer 1

`f(x)=9\cdot 2^x`

Explanation:

The general form of an exponential function is:

`\Large\boxed{y=ab^x}`

where:

• a is the initial value (y-intercept).
• b is the base (growth/decay factor) in decimal form.

To find the formula for an exponential function that passes through points (-1, 9/2) and (3, 72), substitute both points into the formula to create two equations in terms of a and b:

`\textsf{Equation\;1:} \quad ab^(-1)=(9)/(2)`

`\textsf{Equation\;2:} \quad ab^(3)=72`

Divide equation 2 by equation 1 to eliminate a, then solve for b:

`(ab^3)/(ab^(-1))=(72)/((9)/(2))`

`(b^3)/(b^(-1))=(72 \cdot 2)/(9)`

`b^(3-(-1))=(144)/(9)`

`b^4=16`

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

`b=2`

Therefore, the value of b is 2.

Substitute the found value of b into the second equation, and solve for a:

`\begin{aligned}ab^3&=72\\b=2 \implies a(2)^3&=72\\8a&=72\\a&=9\end{aligned}`

Therefore, the value of a is 9.

Therefore, the formula for an exponential function that passes through points (-1, 9/2) and (3, 72) is:

`\large\boxed{f(x)=9\cdot 2^x}`