Question

write an exponential function for a graph that passes through the points (1, 18) and (3, 72). write the function in the form $y = a(b)^x$.

Explanation:

Step1: Substitute (1,18) into formula

Substitute $x=1, y=18$ into $y=a(b)^x$:
$18 = a(b)^1 \implies 18 = ab$

Step2: Substitute (3,72) into formula

Substitute $x=3, y=72$ into $y=a(b)^x$:
$72 = a(b)^3$

Step3: Solve for $b$ using substitution

From Step1, $a = \frac{18}{b}$. Substitute into Step2:
$72 = \frac{18}{b} \times b^3$
Simplify: $72 = 18b^2$
Divide both sides by 18: $b^2 = 4$
Take positive root (exponential base >0): $b = 2$

Step4: Solve for $a$

Substitute $b=2$ into $18=ab$:
$18 = a(2) \implies a = \frac{18}{2} = 9$

Step5: Write final function

Combine $a=9$ and $b=2$ into $y=a(b)^x$

Answer:

$y = 9(2)^x$