Question

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Identify the transformations of f(x) given g(x)= -1/3f(-2x+2)-3​

Answer 1

• reflection over the x-axis
• vertical compression by a factor of 1/3
• vertical translation 3 units down
• reflection over the y-axis (negative coefficient of x)
• horizontal compression by a factor of 1/2
• translation 1 unit to the right (x replaced by x-1)

Explanation:

The various scaling and translation transformations of interest are ...

• g(x) = -f(x) -- reflection over the x-axis
• g(x) = k·f(x) -- vertical scaling by a factor of k (k>1 = expansion)
• g(x) = f(x) +k -- vertical translation by k (k>0 = up)
• g(x) = f(-x) -- reflection over the y-axis
• g(x) = f(x/k) -- horizontal scaling by a factor of k (k>1 = expansion)
• g(x) = f(x -k) -- horizontal translation by k (k>0 = right)

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In terms of the vertical transformations, we have ...

• reflection over the x-axis (leading minus sign)
• vertical compression by a factor of 1/3
• vertical translation 3 units down (-3 added to the function value)

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The argument of the function can be written differently to make it easier to see the horizontal transformations.

`g(x)=-(1)/(3)f\left(-(x-1)/((1)/(2))\right)-3`

This shows you the horizontal transformations are ...

• reflection over the y-axis (negative coefficient of x)
• horizontal compression by a factor of 1/2
• translation 1 unit to the right (x replaced by x-1)