Question

Which option correctly describes the transformation of f(x)=2x+3−−−−√−4
𝑓
(
𝑥
)
=
2
𝑥
+
3
−
4
to g(x)=2x+8−−−−√−4
𝑔
(
𝑥
)
=
2
𝑥
+
8
−
4
?(1 point)
Responses

shift left 8 units
shift left 8 units

shift left 5 units
shift left 5 units

shift up 5 units
shift up 5 units

shift right 5 units

Answer 1

To analyze the transformation from \( f(x) = \sqrt{2x + 3} - 4 \) to \( g(x) = \sqrt{2x + 8} - 4 \), we need to focus on the expression inside the square root:

- The transformation from \( 2x + 3 \) to \( 2x + 8 \) can be seen by rewriting it as:
\[
2x + 3 \quad \text{to} \quad 2x + 8
\]

We can see:
- \( 2x + 3 \) can be rewritten as \( 2x + 5 - 2 \) to get to \( 2x + 8 \).
- This indicates a horizontal shift in relation to the \( x \)-values.

Now let's set it equal:
\[
2x + 3 + 5 = 2x + 8
\]
This indicates we are shifting the graph to the left.

More precisely, it is effectively translating the \( x \)-values:
\[
(2x + 5) = 2(x+5/2)
\]

So the term inside the square root has changed from \( 3 \) to \( 8 \) (adding \( 5 \)). To find out how to achieve this functionally, you can look at the change of the input:
- From 3 to 8 amounts to a shift of \( 5 \), which signifies a leftward transformation.

Thus, the correct choice is:

**shift left 5 units.**