To solve this problem, we need to understand how to transform the function [tex]\( a(x) = 195x \)[/tex] into the function [tex]\( c(x) = 150x + 90 \)[/tex]. These transformations involve changes in both the rate of change (slope) and the constant (y-intercept).
1. Comparing the slopes:
- The slope of [tex]\( a(x) = 195x \)[/tex] is 195.
- The slope of [tex]\( c(x) = 150x + 90 \)[/tex] is 150.
To convert the slope of 195 to 150, we can write the transformed slope as a fraction:
[tex]\[
\frac{150}{195} = \frac{10}{13}
\][/tex]
This means that [tex]\( a(x) \)[/tex] is transformed by multiplying the input by [tex]\( \frac{10}{13} \)[/tex], resulting in a vertical shrink by a factor of [tex]\( \frac{10}{13} \)[/tex].
2. Considering the constant term:
- The function [tex]\( a(x) \)[/tex] has a y-intercept of 0 (i.e., it passes through the origin).
- The function [tex]\( c(x) \)[/tex] has a y-intercept of 90.
Additionally, there is an extra amount added to the function, indicating a vertical translation of 90 units upward.
By combining these two transformations, we need to shrink the graph vertically by a factor of [tex]\( \frac{10}{13} \)[/tex] and then shift it up by 90 units.
Hence, the correct transformation from [tex]\( a(x) \)[/tex] to [tex]\( c(x) \)[/tex] is a vertical shrink by [tex]\( \frac{10}{13} \)[/tex] and then a vertical translation 90 units up.
This confirms that the correct answer is:
- vertical shrink by [tex]\( \frac{10}{13} \)[/tex]
- and then a vertical translation 90 units up.
