To determine the transformations applied to the graph of the function [tex]\( f(x) = 10^x \)[/tex] to obtain the graph of the function [tex]\( g(x) = 3(10^x) - 2 \)[/tex], we can analyze the components of the function [tex]\( g(x) \)[/tex] step-by-step.
1. Vertical Dilation:
- The function [tex]\( f(x) = 10^x \)[/tex] produces a graph where the y-values are powers of 10.
- The term [tex]\( 3(10^x) \)[/tex] indicates that every y-value of [tex]\( f(x) \)[/tex] is multiplied by 3.
- This is a vertical dilation by a factor of 3.
2. Vertical Shift:
- The term [tex]\( -2 \)[/tex] in [tex]\( g(x) = 3(10^x) - 2 \)[/tex] affects the y-values.
- Subtracting 2 from [tex]\( 3(10^x) \)[/tex] shifts the entire graph down by 2 units.
- This is a vertical shift down by 2 units.
Combining both transformations:
- The vertical dilation by a factor of 3 scales the graph vertically.
- The vertical shift down by 2 units moves the entire graph down along the y-axis.
Therefore, the correct transformations applied are a vertical dilation by a factor of 3 and a vertical shift down 2 units.
Thus, the correct answer is:
C. a vertical dilation by a factor of 3 and a vertical shift down 2 units
