Answer:
[tex]f(x)=2(2)^{0.5x}-3[/tex]
Step-by-step explanation:
Parent function:
[tex]g(x)=2^x[/tex]
Properties of the given parent function:
- y-intercept at (0, 1)
- horizontal asymptote at y = 0
- As x → -∞, y → 0
- As x → ∞, y → ∞
Given form of function f(x):
[tex]f(x)=a(b)^{kx}+c[/tex]
If the parent function is [tex]g(x)=2^x[/tex] then b = 2:
[tex]\implies f(x)=a(2)^{kx}+c[/tex]
From inspection of the graphed function f(x):
- y-intercept at (0, -1)
- horizontal asymptote at y = -3
Therefore, the y-intercept has shifted 2 units down, yet the asymptote has shifted 3 units down. This implies that there has been a vertical shift of 3 units down and a vertical stretch.
The vertical shift is denoted by the variable "c" so c = -3:
[tex]\implies f(x)=a(2)^{kx}-3[/tex]
The vertical stretch is denoted by the variable "a". To find value of a, substitute the point of the y-intercept into the equation:
[tex]\begin{aligned}f(0) & = -1\\\implies a(2)^{k \times 0}-3 & =-1\\a-3 & = -1\\a-3+3 & = -1+3\\a & = 2\end{aligned}[/tex]
Therefore, as a = 2:
[tex]\implies f(x)=2(2)^{kx}-3[/tex]
From inspection of the given graph, the curve passes through point (4, 5). Substitute this point into the equation to find the value of k:
[tex]\begin{aligned}f(4) & = 5\\\implies 2(2)^{4k}-3 & =5\\2(2)^{4k}& =8\\(2)^{4k}& =4\\(2)^{4k}& =2^2\\4k & = 2\\k & = 0.5\end{aligned}[/tex]
Therefore, the equation of the function f(x) is:
[tex]\implies f(x)=2(2)^{0.5x}-3[/tex]
