To determine the correct transformations applied to the function [tex]\( f(x) = \sqrt{3x + 3} + 3 \)[/tex], let's analyze the function step-by-step:
### Step 1: Identifying Basic Transformations
We begin with the basic square root function [tex]\( g(x) = \sqrt{x} \)[/tex]. Now, consider the given function [tex]\( f(x) = \sqrt{3x + 3} + 3 \)[/tex].
### Step 2: Analyzing Inside the Square Root
Look at the expression inside the square root: [tex]\(3x + 3\)[/tex].
- Horizontal Compression: The coefficient of [tex]\(x\)[/tex] is [tex]\(3\)[/tex]. To understand this transformation, rewrite [tex]\(3x + 3\)[/tex] as [tex]\(3(x + 1)\)[/tex]. The [tex]\(3\)[/tex] inside the square root suggests that there is a horizontal compression by a factor of [tex]\( \frac{1}{3} \)[/tex]. This means that the function will be compressed horizontally to one-third of its original width.
- Shift Left: Rewriting the term as [tex]\(3(x + 1)\)[/tex] indicates a shift to the left by [tex]\(1\)[/tex] unit because [tex]\(x\)[/tex] is replaced with [tex]\(x + 1\)[/tex].
### Step 3: Analyzing Outside the Square Root
Next, look at the expression outside the square root: [tex]\(+ 3\)[/tex].
- Vertical Shift: The [tex]\(+ 3\)[/tex] outside the square root means that the function is shifted upward by [tex]\(3\)[/tex] units.
### Summary of Transformations
Combining all the above insights, the transformations applied to the function [tex]\( f(x) = \sqrt{3x + 3} + 3 \)[/tex] are:
- A horizontal compression by a factor of [tex]\( \frac{1}{3} \)[/tex]
- A shift to the left by [tex]\( 1 \)[/tex] unit
- A shift upward by [tex]\( 3 \)[/tex] units
Therefore, the correct set of transformations is:
- Horizontal Compression, Left 1, Up 3
Based on the analysis, the correct option is:
- Horizontal Compression, Left 1, Up 3
Thus, the choice should be the one numbered as:
[tex]\[ \boxed{4} \][/tex]
