Answer:
The exponential function is [tex]y=400(4.31)^{t}[/tex]
Step-by-step explanation:
* Lets look to the graph and describe it
- The graph is an exponential function function
- It is a relation between t and y
- t is represented by x-axis
- Y is represented by y-axis
- The vale of y is 400 when t = 0
- The value of y is 32,000 when t = 3
* We have two point on the graph so we can make the equation
- The form of the exponential function is [tex]y=a(b)^{t}[/tex], where
a is the initial value and b is the growth factor
∵ The graph of the function is growth exponential function
∴ b is greater than 1
∵ The points (0 , 400) and (3 , 32000) lie on the function
∵ y = 400 when t = 0
∴ the initial value is 400
∴ a = 400
∴ [tex]y=400(b)^{t}[/tex]
- Substitute the values of y and t by the second point
∴ [tex]32000=400(b)^{3}[/tex]
- Divide both sides by 400
∴ [tex]80=(b)^{3}[/tex]
- Take ∛ for both sides
∴ ∛80 = b
∴ b = 4.31
∴ [tex]y=400(4.31)^{t}[/tex]
* The exponential function is [tex]y=400(4.31)^{t}[/tex]
The equation of the given exponential function can be written as [tex]y=400(1.7^x)[/tex].
The general equation of an exponential function is written as,
[tex]y=a(b^x)[/tex]
Where (x,y) are the coordinates, and a and b are the constant values.
We know that the graph is a function of an exponential function, therefore,
[tex]y=a(b^x)[/tex]
since, there are two points on the graph that are already defined we will use those points to find the value of a and b,
point 1, (0,400)
point 2, (3, 2000)
Substitute the value of point 1,
[tex]y=a(b^x)\\\\400 = a(b^0)\\\\400=a[/tex]
Substitute the value of point 2 and 'a' in the equation,
[tex]y=a(b^x)\\\\2000 = 400(b^3)\\\\b^3 = 5\\\\b=\sqrt[3]{5}\\\\b = 1.7[/tex]
Hence, the equation of the given exponential function can be written as [tex]y=400(1.7^x)[/tex].
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