Question

Write an exponential function in the form y = ab” that goes through points (0,18)and (3,9216).​

Answer 1

Answer:

$y = 18 \cdot \left(8^x\right)$

Step-by-step explanation:

The question is asking for an exponential function in the form $y = a\cdot b^x$. The goal is to find the value of $a$ and $b$ such that both $(0,\, 18)$ and $(3,\, 9216)$ are on the graph of this function.

Saying that a point $(m,\, n)$ is on this function is the same as stating that if $x = m$, then $y = a \cdot b^m = n$.

• $(0,\, 18)$ is on $y = a\cdot b^x$. Therefore, when $x = 0$, it must be true that $y = a \cdot b^0 = 18$. On the other hand, any non-zero number to the $0$th power is $1$. Since (apparently) $b \ne 0$, $a \cdot b^0 = a$. Conclusion: $a = a \cdot b^0 = 18$.

• $(3,\, 9216)$ is on $y = a\cdot b^x$. Therefore, when $x = 3$, it must be true that $y = a \cdot b^{3} = 9216$. Since it is determined that $a = 18$, $18\, b^3 = 9216$. Hence $b^3 = 9216 / 18 = 512$. Take the cube-root of both sides: $b = \sqrt[3]{512} = 8$.

Conclusion:

• $a = 18$.
• $b = 8$.

Therefore, $y = 18\cdot \left(8^x\right)$.