Question

Analyze the graphs g(x)=−4x+1−−−−√+3.5
𝑔
(
𝑥
)
=
−
4
𝑥
+
1
+
3.5
and h(x)=−4x+1−−−−√−4.5
ℎ
(
𝑥
)
=
−
4
𝑥
+
1
−
4.5
. Which option correctly describes the transformation to take g(x)
𝑔
(
𝑥
)
to h(x)
ℎ
(
𝑥
)
? (1 point)
Responses

vertical translation up 8 units
vertical translation up 8 units

vertical translation down 7 units
vertical translation down 7 units

vertical translation down 8 units
vertical translation down 8 units

vertical translation right 8 units

Answer 1

To analyze the transformation from \( g(x) \) to \( h(x) \), we first write out both functions clearly:

1. \( g(x) = -4\sqrt{x+1} + 3.5 \)
2. \( h(x) = -4\sqrt{x+1} - 4.5 \)

Now, we can compare the two functions:

- The main difference between \( g(x) \) and \( h(x) \) is in the constant term.

This can be expressed as:
- \( g(x) \) has a constant term of \( +3.5 \).
- \( h(x) \) has a constant term of \( -4.5 \).

To find the vertical translation, we can calculate the difference in the constant terms:

\[
k = -4.5 - 3.5 = -8
\]

This means \( h(x) \) is obtained by taking \( g(x) \) and translating it **downward by 8 units**.

Thus, the correct response is:

**vertical translation down 8 units**.