Questions: Given the function f(x)=log5(x), which of the following functions is a transformation of f(x) right 6 units, up 3 units, stretched vertically by a factor of 2, and reflected across the x axis? (1 point) g(x)=-1/2 log5(x+6)-3 g(x)=2 log5(-x+6)+3 g(x)=-2 log5(x-6)+3 g(x)=-log5(1/2 x-6)-3

Given the function f(x)=log5(x), which of the following functions is a transformation of f(x) right 6 units, up 3 units, stretched vertically by a factor of 2, and reflected across the x axis? (1 point) g(x)=-1/2 log5(x+6)-3 g(x)=2 log5(-x+6)+3 g(x)=-2 log5(x-6)+3 g(x)=-log5(1/2 x-6)-3
Transcript text: Given the function $f(x)=\log _{5}(x)$, which of the following functions is a transformation of $f(x)$ right 6 units, up 3 units, stretched vertically by a factor of 2 , and reflected across the $x$ axis? (1 point) $g(x)=-\frac{1}{2} \log _{5}(x+6)-3$ $g(x)=2 \log _{5}(-x+6)+3$ $g(x)=-2 \log _{5}(x-6)+3$ $g(x)=-\log _{5}\left(\frac{1}{2} x-6\right)-3$
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Solution

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Solution Steps

Solution Approach

To transform the function \( f(x) = \log_{5}(x) \) as described:

  1. Right 6 units: Replace \( x \) with \( x - 6 \).
  2. Up 3 units: Add 3 to the function.
  3. Stretched vertically by a factor of 2: Multiply the function by 2.
  4. Reflected across the x-axis: Multiply the function by -1.

The transformed function is \( g(x) = -2 \log_{5}(x - 6) + 3 \).

Step 1: Define the Original Function

The original function is given by

\[ f(x) = \log_{5}(x) \]

Step 2: Apply the Transformations

To transform \( f(x) \) according to the specified operations:

  1. Right 6 units: Replace \( x \) with \( x - 6 \), resulting in \( \log_{5}(x - 6) \).
  2. Stretched vertically by a factor of 2: Multiply the function by 2, yielding \( 2 \log_{5}(x - 6) \).
  3. Reflected across the x-axis: Multiply the entire function by -1, giving \( -2 \log_{5}(x - 6) \).
  4. Up 3 units: Add 3 to the function, resulting in

\[ g(x) = -2 \log_{5}(x - 6) + 3 \]

Step 3: Final Form of the Transformed Function

The final transformed function is

\[ g(x) = -\frac{2 \log(x - 6)}{\log(5)} + 3 \]

Final Answer

The transformed function is

\[ \boxed{g(x) = -2 \log_{5}(x - 6) + 3} \]

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