Describe the translation of ƒ(x) = √x (green) to g(x) (blue)

Question

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
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1 point
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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.