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Given the function f(x)=log5 x

, which of the following functions is the 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)
Responses

g(x)=−1/2log5(x+6)−3
g of x is equal to negative 1 half log sub 5 open paren x plus 6 close paren minus 3

g(x)=−2 log5(x−6)+3
g of x is equal to negative 2 log sub 5 open paren x minus 6 close paren plus 3

g(x)=−log5(1/2x−6)−3
g of x is equal to negative log sub 5 times open paren 1 half x minus 6 close paren minus 3

g(x)=2 log5(−x+6)+3

1 answer

To transform the function \( f(x) = \log_5 x \) according to the given specifications, we need to perform the following steps:

  1. Right 6 units: This transformation shifts the graph of the function to the right by 6 units, resulting in \( f(x) \) becoming \( f(x-6) = \log_5(x-6) \).

  2. Up 3 units: This transformation raises the entire graph by 3 units, which can be expressed as \( f(x-6) + 3 = \log_5(x-6) + 3 \).

  3. Stretch vertically by a factor of 2: This transformation scales the function vertically by 2, giving us \( 2[f(x-6) + 3] = 2\log_5(x-6) + 6 \).

  4. Reflect across the x-axis: This transformation negates the entire expression, resulting in \( -[2\log_5(x-6) + 6] = -2\log_5(x-6) - 6 \).

Now to summarize the transformation steps:

  • Start with \( f(x) = \log_5(x) \)
  • Shift right 6 units: \( \log_5(x-6) \)
  • Shift up 3 units: \( \log_5(x-6) + 3 \)
  • Stretch vertically by a factor of 2: \( 2\log_5(x-6) + 6 \)
  • Reflect across the x-axis: \( -[2\log_5(x-6) + 6] = -2\log_5(x-6) - 6 \)

Thus, the final transformation is:

\[ g(x) = -2\log_5(x - 6) - 6 \]

Now, let's check the options provided:

  1. \( g(x) = -\frac{1}{2}\log_5(x + 6) - 3 \)
  2. \( g(x) = -2 \log_5(x - 6) + 3 \)
  3. \( g(x) = -\log_5\left(\frac{1}{2}x - 6\right) - 3 \)
  4. \( g(x) = 2 \log_5(-x + 6) + 3 \)

None of the listed options seem to match the derived transformation \( -2\log_5(x - 6) - 6 \).

However, upon closely reviewing option 2: \( g(x) = -2 \log_5(x - 6) + 3 \), it appears to have an error in the vertical shift amount at the end.

Thus, it seems the closest option that performs all the transformations as specified (except for the final vertical shift which should be adjusted to -6 instead of +3) is:

Option 2: \( g(x) = -2 \log_5(x - 6) + 3 \).

Please note there appears to be a mistake in the choices as they do not reflect precisely the required transformations.