Answer:
[tex]y=4(1.5)^x[/tex]
Step-by-step explanation:
General form of an exponential function: [tex]y=ab^x[/tex]
where:
- a is the y-intercept (or initial value)
- b is the base (or growth factor)
- x is the independent variable
- y is the dependent variable
If b > 1 then it is an increasing function
If 0 < b < 1 then it is a decreasing function
Given ordered pairs:
(3, 13.5) and (5, 30.375)
As the y-values are increasing, the function is increasing, so b > 1
Input the given ordered pairs into the general form of the equation:
[tex]\implies ab^3=13.5[/tex]
[tex]\implies ab^5=30.375[/tex]
To find b, divide the second equation by the first:
[tex]\implies \dfrac{ab^5}{ab^3}=\dfrac{30.375}{13.5}[/tex]
[tex]\implies b^2=2.25[/tex]
[tex]\implies b=\pm \sqrt{2.25}[/tex]
[tex]\implies b= \pm 1.5[/tex]
As the function is increasing, b > 1:
⇒ b = 1.5 only
Substitute the found value of b into one of the equations and solve for a:
[tex]\implies a(1.5)^3=13.5[/tex]
[tex]\implies 3.375a=13.5[/tex]
[tex]\implies a=\dfrac{13.5}{3.375}[/tex]
[tex]\implies a=4[/tex]
Therefore, the final exponential equation is:
[tex]y=4(1.5)^x[/tex]
