To transform the parent function [tex]\( f(x) = 3^x \)[/tex] into the function [tex]\( g(x) = -2 \cdot 3^{-2x + 2} - 1 \)[/tex], we need to apply a series of transformations. Below are the transformations stated in the correct order along with the terminology used:
1. Reflection across the y-axis and horizontal compression by a factor of 2:
- This step involves replacing [tex]\( x \)[/tex] with [tex]\( -2x \)[/tex] in the exponent of the parent function. This transformation mirrors the graph across the y-axis (reflection) and then compresses it horizontally by a factor of 2.
- The resulting function after this transformation is [tex]\( 3^{-2x} \)[/tex].
2. Horizontal shift to the left by 1 unit:
- To achieve this, we replace [tex]\( -2x \)[/tex] with [tex]\( -2(x - 1) \)[/tex] which equivalently adds 2 in the exponent: [tex]\( 3^{-2x+2} = 3^{-2(x-1)} \)[/tex].
- This transformation moves the graph 1 unit to the left.
- The resulting function after this transformation is [tex]\( 3^{-2(x - 1)} \)[/tex].
3. Vertical stretch by a factor of 2 and reflection across the x-axis:
- This step involves multiplying the entire function by [tex]\(-2\)[/tex]. The multiplication by 2 stretches the graph vertically by a factor of 2, and the negative sign reflects the graph across the x-axis.
- The resulting function after this transformation is [tex]\( -2 \cdot 3^{-2(x - 1)} \)[/tex].
4. Vertical shift downward by 1 unit:
- Finally, we subtract 1 from the function, shifting the graph downward by 1 unit.
- The resulting function after this final transformation is [tex]\( -2 \cdot 3^{-2(x - 1)} - 1 \)[/tex].
To summarize, the transformations applied in order are:
1. Reflection across the y-axis and horizontal compression by a factor of 2
2. Horizontal shift to the left by 1 unit
3. Vertical stretch by a factor of 2 and reflection across the x-axis
4. Vertical shift downward by 1 unit
