Question

Find an exponential function that passes through each pair of points.(2,1.75) & (-2,28)

Answer 1

Answer:

$f(x) = 7\, ((1/2)^{x})$.

Step-by-step explanation:

An exponential function is typically in the form $f(x) = a\, (b^{x})$, where $a$ and $b$ ($b > 0$) are constants to be found.

In this question:

$f(2) = 1.75$ means that $a\, (b^{2}) = 1.75$.

$f(-2) = 28$ means that $a\, (b^{-2}) = 28$.

Divide one of the two equations by the other to eliminate $a$ and solve for $b$.

The number of the right-hand side of the second equation is larger than that of the first equation. Hence, divide the second equation with the first:

$\displaystyle \frac{a\, (b^{-2})}{a\, (b^{2})} = \frac{28}{1.75}$.

$\displaystyle b^{-4} = 16$.

$b^{-1} = 2$.

$\displaystyle b = \frac{1}{2}$.

Substitute $b = (1/2)$ back into either equation (for example, the first equation) and solve for $a$:

$a\, ((1/2)^{2}) = 1.75$.

$a = 7$.

Substitute $a = 7$ and $b = (1/2)$ into the other equation. That equation should also be satisfied.

Therefore, this function would be:

$f = 7\, ((1 / 2)^{x})$.