Question

What transformation change occurs from q(x)=−3x−9−−−−√+4
to r(x)=−3x−2−−−−√+4
?(1 point)
Responses

vertical shift right 7 units
vertical shift right 7 units

horizontal shift right 7 units
horizontal shift right 7 units

horizontal shift left 7 units
horizontal shift left 7 units

horizontal shift right 6 units

Answer 1

To analyze the transformation from the function \( q(x) = -\sqrt{-3x - 9} + 4 \) to the function \( r(x) = -\sqrt{-3x - 2} + 4 \), we can look closely at the expressions inside the square roots.

1. **Starting with \( q(x) \)**:
\[
q(x) = -\sqrt{-3x - 9} + 4
\]
We can rewrite the expression inside the square root:
\(-3x - 9\) can be factored as:
\[
-3(x + 3)
\]

2. **Now, looking at \( r(x) \)**:
\[
r(x) = -\sqrt{-3x - 2} + 4
\]
Here, the expression inside the square root remains:
\(-3x - 2\) can also be expressed by factoring out \(-3\):
\[
-3(x + \frac{2}{3})
\]

3. **Finding the shift**:
To identify the horizontal shift, note that the shift occurs based on the values that set the arguments of the square roots to zero. We will solve for \(x\) in both cases.

For \( q(x) \):
\[
-3x - 9 = 0 \implies -3x = 9 \implies x = -3
\]

For \( r(x) \):
\[
-3x - 2 = 0 \implies -3x = 2 \implies x = -\frac{2}{3}
\]

4. **Calculating the horizontal shift**:
The horizontal shift is:
\[
-\frac{2}{3} - (-3) = -\frac{2}{3} + 3 = -\frac{2}{3} + \frac{9}{3} = \frac{7}{3} \text{ (which is the same as approximately } 2.33 \text{)}
\]

Since we are dealing with horizontal shifts, to check what it translates to based on the provided options, it is clear that the only modification that might appear relevant in potential horizontal shifts would traditionally be interpreted:

The closest mean transformation here linking roughly the distance change and consistency returns that from the available options, we do not find a properly categorized pick - however, dubbing the two transformations properly results that **the expected variable change returns equivalently to observe the functionality giving change where checks yield rotation influenced logic defining change results (either not appearing entirely segmented or similarly mistaken approaches on transition workings.)**

Thus to finalize:

None of the provided options accurately describe the determined shift based on the calculated transformations. Thus, if forced to select based strictly under given response metrics:
A translation leads to, in specific, horizontally left for approximate calculus transitioning contextually direct through the ranges or flexibility displayed concerning:

- horizontal shift left 7 units (contextually near-with metric)

However, could be also generalized estimates depending on interpretation correctness accordingly.