Given the function, [tex]$f(x)=\sqrt{3x+3}-3$[/tex], choose the correct transformation(s).
A. Horizontal Compression, Left 1, Down 3 B. Vertical Stretch, Left 3, Down 3 C. Vertical Compression, Left 1, Down 3 D. Horizontal Stretch, Left 3, Down 3
To analyze the transformations of the given function [tex]\( f(x) = \sqrt{3x + 3} - 3 \)[/tex], we will look at each component of the function and determine the specific transformations applied to the base function, which is [tex]\( \sqrt{x} \)[/tex].
### Step 1: Horizontal and Vertical Shifts, Scaling 1. Inside the square root function ([tex]\( \sqrt{3x + 3} \)[/tex]): - The [tex]\( 3x \)[/tex] component suggests a horizontal compression by a factor of [tex]\( \frac{1}{3} \)[/tex]. - The [tex]\( +3 \)[/tex] inside the square root function indicates a shift to the left by 1 unit. This comes from solving [tex]\( 3x + 3 = 0 \)[/tex] which gives [tex]\( x = -1 \)[/tex].
2. Outside the square root function ([tex]\( -3 \)[/tex]): - The [tex]\( -3 \)[/tex] outside the square root function indicates a vertical shift downward by 3 units.
### Summary of Transformations: - Horizontal Compression by a factor of [tex]\( \frac{1}{3} \)[/tex] - Shift to the left by 1 unit - Shift downward by 3 units
Therefore, the correct set of transformations for the function [tex]\( f(x) = \sqrt{3x + 3} - 3 \)[/tex] is:
- Horizontal Compression by a factor of [tex]\( \frac{1}{3} \)[/tex] - Shift to the left by 1 unit - Shift downward by 3 units
Hence, the transformations described as: Vertical Compression, Left 1, Down 3, accurately reflect the changes made to the base function [tex]\( \sqrt{x} \)[/tex].
