We need to determine which transformations must be applied to function [tex]\( f(x) = 12^x \)[/tex] to produce the graph of function [tex]\( g(x) = -2(12)^x \)[/tex].
1. Reflection over the [tex]\( x \)[/tex]-axis:
- The function [tex]\( g(x) = -2(12)^x \)[/tex] includes a negative sign, indicating that each value of the function is reflected over the [tex]\( x \)[/tex]-axis. This changes the sign of the output values of the function.
2. Vertical stretch:
- The coefficient [tex]\( -2 \)[/tex] in [tex]\( g(x) \)[/tex] indicates a vertical stretch. Specifically, the graph of [tex]\( 12^x \)[/tex] is scaled by a factor of 2. Normally, if the factor is more than 1, it results in a vertical stretch.
Therefore, the transformations needed to convert [tex]\( f(x) = 12^x \)[/tex] to [tex]\( g(x) = -2(12)^x \)[/tex] are:
- Reflection over the [tex]\( x \)[/tex]-axis: This is due to the negative sign in front of the function.
- Vertical stretch: This is due to the multiplication by 2.
Hence, the correct transformations are "reflection over the [tex]\( x \)[/tex]-axis" and "vertical stretch." No vertical shift, horizontal shift, or vertical compression is required.
