Math Problem Statement

describe the transformations from the parent graph g(x)=-square root of x-3+2

Solution

The given function is g(x)=x3+2g(x) = -\sqrt{x - 3} + 2. To describe the transformations from the parent graph f(x)=xf(x) = \sqrt{x}, we break down the changes step by step:

  1. Horizontal Translation (Right Shift):

    • The term inside the square root, (x3)(x - 3), indicates a horizontal shift.
    • Since it's x3x - 3, the graph shifts 3 units to the right.
  2. Reflection (Vertical Flip):

    • The negative sign in front of the square root, x3-\sqrt{x - 3}, reflects the graph across the x-axis. This means the graph will now point downward instead of upward.
  3. Vertical Translation (Up Shift):

    • The +2+2 outside the square root shifts the graph 2 units up.

Summary of Transformations:

  • The parent graph f(x)=xf(x) = \sqrt{x} is first shifted 3 units to the right.
  • It is then reflected across the x-axis (flipped upside down).
  • Finally, it is shifted 2 units up.

Would you like more details on any of these transformations?

5 Related Questions:

  1. How would the graph change if the function was g(x)=x+3g(x) = \sqrt{x + 3}?
  2. What does a vertical stretch or compression look like in a square root function?
  3. How can you graph g(x)=x3+2g(x) = -\sqrt{x - 3} + 2 by hand?
  4. What effect does a horizontal reflection (over the y-axis) have on the square root function?
  5. How do horizontal translations differ from vertical translations in general?

Tip:

When analyzing transformations, always start with the operations inside the function (like shifts and reflections), then move to outside operations (like stretching and translations).

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Math Problem Analysis

Mathematical Concepts

Graph Transformations
Square Root Functions
Function Reflections
Translations

Formulas

g(x) = -√(x - 3) + 2
f(x) = √x

Theorems

Transformation of Functions Theorem

Suitable Grade Level

Grades 9-12