f(x)=square root x was transformed to create the function in the graph. Interpret the key features of the graph. What type of transformation occurred from the parent function

The given function \( f(x) = \sqrt{x} - 2 \) is derived from the parent function \( f(x) = \sqrt{x} \) by applying a vertical transformation. Let’s break down the key features and transformations that are occurring here:

Parent Function

The parent function \( f(x) = \sqrt{x} \) has the following characteristics:

1. Domain: \( x \geq 0 \)
2. Range: \( y \geq 0 \)
3. Key Points: Starts at the origin (0,0), with points like (1,1), (4,2), (9,3), etc.

Transformation

The transformation from \( f(x) = \sqrt{x} \) to \( f(x) = \sqrt{x} - 2 \) can be understood as follows:

1. Vertical Shift Downward: The graph of the function \( f(x) = \sqrt{x} - 2 \) represents a downward shift of the entire graph of \( f(x) = \sqrt{x} \) by 2 units. This means that every point on the parent function will be moved down by 2 units.

Key Features of \( f(x) = \sqrt{x} - 2 \)

• Domain: The domain remains the same as the parent function: \( x \geq 0 \).
• Range: The range is shifted down by 2 units, so it becomes \( y \geq -2 \).
• New Key Points:
  • Starting point: The original point (0,0) becomes (0, -2).
  • Using other points on the parent function:
    • For \( x = 1 \): \( f(1) = \sqrt{1} - 2 = -1 \) ➔ (1, -1)
    • For \( x = 4 \): \( f(4) = \sqrt{4} - 2 = 0 \) ➔ (4, 0)
    • For \( x = 9 \): \( f(9) = \sqrt{9} - 2 = 1 \) ➔ (9, 1)

Points Provided

The points provided in the graph are:

• (2, 0)
• (3, 1)
• (6, 2)

Let's check if these points adhere to the function \( f(x) = \sqrt{x} - 2 \):

1. For \( x = 2 \): \[ f(2) = \sqrt{2} - 2 \approx 1.414 - 2 = -0.586 \quad \text{(Does not match (2,0))} \]
2. For \( x = 3 \): \[ f(3) = \sqrt{3} - 2 \approx 1.732 - 2 = -0.268 \quad \text{(Does not match (3,1))} \]
3. For \( x = 6 \): \[ f(6) = \sqrt{6} - 2 \approx 2.449 - 2 = 0.449 \quad \text{(Does not match (6,2))} \]

It seems that the provided points do not actually follow from the transformation of the square root function as described by \( f(x) = \sqrt{x} - 2 \).

Conclusion

• The transformation from \( f(x) = \sqrt{x} \) to \( f(x) = \sqrt{x} - 2 \) involved a vertical shift downward by 2 units.
• The key features:
  • Domain: \( x \geq 0 \)
  • Range: \( y \geq -2 \)
• The provided points do not fit the transformed function accurately, indicating a potential error in the suggested points or perhaps a different transformation or function.