Answer:
The required function is [tex]g(x)=\frac{1}{2}(5^{x-3})-2[/tex].
Step-by-step explanation:
The given parent function is
[tex]f(x)=5^x[/tex]
The transformation of a parent function is defined as
[tex]g(x)=kf(x+a)+b[/tex]
Where, k represents the vertical stretch or compression, a represents the horizontal shift and b represent the vertical shift.
→ If |k|>1, then it represents vertically stretch and If |k|<1, then it represent vertically compression.
→ If a>0, then f(x) shifts left by a units and If a<0, then f(x) shifts right by a units.
→If b>0, then f(x) shifts upward by b units and If b<0, then f(x) shifts downward by b units.
Since the graph of f(x) has been vertically compressed by a factor of one-half shifted to the right three units and down two units.
[tex]k=\frac{1}{2}[/tex]
[tex]a=-3[/tex]
[tex]b=-2[/tex]
The graph of required function is
[tex]g(x)=\frac{1}{2}f(x-3)-2[/tex]
[tex]g(x)=\frac{1}{2}(5^{x-3})-2[/tex]
Therefore the required function is [tex]g(x)=\frac{1}{2}(5^{x-3})-2[/tex].
Function transformation involves changing the form of a function
The function that represents the transformation is: g(x) = 1/2(5^(x -3))
The parent function is given as:
f(x) = 5^x
When the function is vertically compressed by a factor of one-half, the function becomes
f'(x) = 1/2(5^x)
When the function is shifted 3 units right, it becomes
f"(x) = 1/2(5^(x -3))
Rewrite the function as:
g(x) = 1/2(5^(x -3))
Hence, the function that represents the transformation is: g(x) = 1/2(5^(x -3))
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