Question

Which function represents a vertical stretch by a factor of 1.85 with a reflection over the line y=−1
𝑦
=
−
1
on function w(x)=−3(x+2)−−−−−−−√−1
𝑤
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𝑥
)
=
−
3
(
𝑥
+
2
)
−
1
?(1 point)
Responses

k(x)=4.85(x+2)−−−−−−−−−√−1
𝑘
(
𝑥
)
=
4.85
(
𝑥
+
2
)
−
1
k left parenthesis x right parenthesis equals Start Root 4.85 left parenthesis x plus 2 right parenthesis End Root minus 1

h(x)=3(x+2)√1.85−1
ℎ
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𝑥
)
=
3
(
𝑥
+
2
)
1.85
−
1
h left parenthesis x right parenthesis equals Start Fraction Start Root 3 left parenthesis x plus 2 right parenthesis End Root over 1.85 End Fraction minus 1

r(x)=1.853(x+2)−−−−−−−√−1
𝑟
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𝑥
)
=
1.85
3
(
𝑥
+
2
)
−
1
r left parenthesis x right parenthesis equals 1.85 Start Root 3 left parenthesis x plus 2 right parenthesis End Root minus 1

t(x)=1.85+3(x+2)−−−−−−−√−1

Answer 1

To find the function that represents a vertical stretch by a factor of \(1.85\) with a reflection over the line \(y = -1\) for the function \(w(x) = -3\sqrt{x + 2} - 1\), we need to apply both transformations step by step.

1. **Reflection Over the Line \(y = -1\)**:
To reflect a function \(f(x)\) over the line \(y = -1\), we can do this in two steps:
- First, we reflect it over the x-axis: this transforms \(f(x)\) into \(-f(x)\).
- Then, we shift it down by 2 units to achieve the reflection over \(y = -1\).

For our function:
\[
w(x) = -3\sqrt{x + 2} - 1
\]
Reflecting first over the x-axis gives:
\[
-w(x) = 3\sqrt{x + 2} + 1
\]
Now, reflecting this over the line \(y = -1\) means we replace \(y\) with \(-1 - (y + 1)\):
\[
3\sqrt{x + 2} + 1 \rightarrow -(3\sqrt{x + 2} + 1) - 1 = -3\sqrt{x + 2} - 2
\]
However, the correct simplification leads to \(3\sqrt{x + 2} - 1\).

2. **Vertical Stretch by a Factor of \(1.85\)**:
To vertically stretch the function by a factor of \(1.85\), we multiply the entire function by \(1.85\):
\[
y = 1.85(-3\sqrt{x + 2} - 2)
\]
However, reflecting the base function over \(y = -1\) would lead us to keep it in the form of \(y = 3\sqrt{x + 2} - 1\). Hence, multiplying \(3\sqrt{x + 2}\) by \(1.85\):
\[
k(x) = 1.85 \cdot 3\sqrt{x + 2} - 1 = 5.55\sqrt{x + 2} - 1
\]

After analyzing the options provided for \(k(x)\), \(h(x)\), \(r(x)\), and \(t(x)\), we find:

- \(k(x) = 4.85\sqrt{x + 2} - 1\): This does not match.
- \(h(x) = 3(x + 2)\sqrt{1.85} - 1\): This does not match correctly as our input format is wrong.
- \(r(x) = 1.85 \cdot 3\sqrt{x + 2} - 1\): This option seems to align directly.
- \(t(x) = 1.85 + 3\sqrt{x + 2} - 1\): Incorrect structure.

Thus, it seems the appropriate function achieved through the transformations is given by:

\[
\boxed{r(x) = 1.85 \cdot 3\sqrt{x + 2} - 1}
\]