Math Problem Statement

write down the transformations implied by sqrt(x+2)+3. find the x transformation, y transformation, and the parent function

Solution

The function f(x)=x+2+3f(x) = \sqrt{x + 2} + 3 can be analyzed in terms of transformations applied to its parent function.

1. Parent Function:

The parent function here is x\sqrt{x}.

2. Transformations:

  • X-Transformation (Horizontal Shift):

    • The term x+2x + 2 indicates a horizontal shift.
    • Since the expression is x+2x + 2, it implies a left shift by 2 units.
    • Transformation: x=x(2)=x+2x' = x - (-2) = x + 2.
  • Y-Transformation (Vertical Shift):

    • The term +3+3 outside the square root indicates a vertical shift.
    • It implies a shift upwards by 3 units.
    • Transformation: y=y+3y' = y + 3.

Summary:

  • Parent Function: x\sqrt{x}
  • Horizontal Shift: Left by 2 units.
  • Vertical Shift: Up by 3 units.

This means the function f(x)=x+2+3f(x) = \sqrt{x + 2} + 3 is obtained by shifting the graph of the parent function x\sqrt{x} 2 units to the left and 3 units up.

Would you like a detailed graph of this transformation or have any further questions?


5 Related Questions:

  1. How does the graph of x\sqrt{x} compare to x+2\sqrt{x+2}?
  2. What happens to the domain and range of the function after the transformation?
  3. How would the function change if the transformation was f(x)=x2+3f(x) = \sqrt{x - 2} + 3?
  4. Can you derive the inverse function of f(x)=x+2+3f(x) = \sqrt{x + 2} + 3?
  5. What would be the effect of a vertical stretch by a factor of 2 on the function?

Tip:

When dealing with horizontal shifts, remember that adding a value inside the function moves the graph to the left, while subtracting moves it to the right.

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Math Problem Analysis

Mathematical Concepts

Algebra
Function Transformations
Square Root Functions

Formulas

f(x) = sqrt(x + 2) + 3
Horizontal shift: x' = x - (-2)
Vertical shift: y' = y + 3

Theorems

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Suitable Grade Level

Grades 9-11