Question

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

horizontal shift left 7 units
horizontal shift left 7 units

vertical shift right 7 units
vertical shift right 7 units

horizontal shift right 7 units
horizontal shift right 7 units

horizontal shift right 6 units

Answer 1

To analyze the transformation from \( q(x) = -\sqrt{-3x - 9} + 4 \) to \( r(x) = -\sqrt{-3x - 2} + 4 \), we first rewrite the expressions in a more convenient form.

Starting with \( q(x) \):
\[
q(x) = -\sqrt{-3(x + 3)} + 4
\]
This indicates the function is shifted to the left by 3 units because of the \( x + 3 \) term inside the square root.

Now, looking at \( r(x) \):
\[
r(x) = -\sqrt{-3(x + \frac{2}{3})} + 4
\]
Here it indicates a transformation on the \(x\) term: we see that \(-3x - 2\) can be factored as \(-3(x + \frac{2}{3})\).

### Now comparing the two transformations:
- For \( q(x) \), the change inside the square root \( -3x - 9 \) implies a shift of 3 units left (because of \( -9\)).
- For \( r(x) \), the expression \( -3(x + \frac{2}{3}) \) suggests a shift left by \( \frac{2}{3} \).

To find the total shift from \( q(x) \) to \( r(x) \):
- From \( -9 \) to \( -2 \), this is an increase of 7 (since \(-2 - (-9) = 7\)).

This means there is a horizontal shift right by 7 units as it goes from \( q(x) \) to \( r(x) \).

### Final Answer:
**horizontal shift right 7 units**