Answer:
[tex]y = 18 \cdot \left(8^x\right)[/tex]
Step-by-step explanation:
The question is asking for an exponential function in the form [tex]y = a\cdot b^x[/tex]. The goal is to find the value of [tex]a[/tex] and [tex]b[/tex] such that both [tex](0,\, 18)[/tex] and [tex](3,\, 9216)[/tex] are on the graph of this function.
Saying that a point [tex](m,\, n)[/tex] is on this function is the same as stating that if [tex]x = m[/tex], then [tex]y = a \cdot b^m = n[/tex].
- [tex](0,\, 18)[/tex] is on [tex]y = a\cdot b^x[/tex]. Therefore, when [tex]x = 0[/tex], it must be true that [tex]y = a \cdot b^0 = 18[/tex]. On the other hand, any non-zero number to the [tex]0[/tex]th power is [tex]1[/tex]. Since (apparently) [tex]b \ne 0[/tex], [tex]a \cdot b^0 = a[/tex]. Conclusion: [tex]a = a \cdot b^0 = 18[/tex].
- [tex](3,\, 9216)[/tex] is on [tex]y = a\cdot b^x[/tex]. Therefore, when [tex]x = 3[/tex], it must be true that [tex]y = a \cdot b^{3} = 9216[/tex]. Since it is determined that [tex]a = 18[/tex], [tex]18\, b^3 = 9216[/tex]. Hence [tex]b^3 = 9216 / 18 = 512[/tex]. Take the cube-root of both sides: [tex]b = \sqrt[3]{512} = 8[/tex].
Conclusion:
- [tex]a = 18[/tex].
- [tex]b = 8[/tex].
Therefore, [tex]y = 18\cdot \left(8^x\right)[/tex].
