Question

Describe the translation of ƒ(x) = √x (green) to g(x) (blue)
A horizontal shift is a translation moves along the x-axis. **To find the direction of the translation, set the transformation expression (within parenthesis) equal to zero and solve. The result will always give your the magnitude and direction of the translation
*
1 point
Captionless Image
g(x) = √(x+2)
g(x) = √(x-2)

Answer 1

To determine the translation from the function \( f(x) = \sqrt{x} \) (green) to \( g(x) \) (blue), we'll analyze the transformations of the function.

Given the general form of a translation for square root functions, we can express it like this:

1. Original function: \( f(x) = \sqrt{x} \)
2. Transformed function: \( g(x) = \sqrt{x + c} \)

Here, the value of \( c \) indicates a horizontal shift:
- If \( c > 0 \), the graph shifts to the left by \( c \) units.
- If \( c < 0 \), the graph shifts to the right by the absolute value of \( c \) units.

Now, let's examine the provided options for \( g(x) \):
- Option 1: \( g(x) = \sqrt{x+2} \)
- Option 2: \( g(x) = \sqrt{x-2} \)

### Step 1: Find the Value of c
1. For \( g(x) = \sqrt{x+2} \):
- Set the expression inside the square root to zero:
\[
x + 2 = 0 \implies x = -2
\]
- Since \( c = +2 \), this indicates a translation **2 units to the left**.

2. For \( g(x) = \sqrt{x-2} \):
- Set the expression inside the square root to zero:
\[
x - 2 = 0 \implies x = 2
\]
- Here, \( c = -2 \), which indicates a translation **2 units to the right**.

### Conclusion
- **If \( g(x) = \sqrt{x + 2} \)**, the translation from \( f(x) = \sqrt{x} \) is **2 units to the left**.
- **If \( g(x) = \sqrt{x - 2} \)**, the translation is **2 units to the right**.

Given this analysis, you can identify the correct transformation based on the context or the specific function you are working with.