Math Problem Statement

How can we transform the solid black function y = √x to match the dotted function shown in the graph?

Solution

To transform the solid black function y=xy = \sqrt{x} to match the dotted function shown in the graph, we need to analyze the transformation applied. This involves horizontal shifts, vertical shifts, reflections, or compressions/stretching.

Observing the Graph:

  1. Horizontal Shift: The dotted function appears shifted to the left compared to the solid black function.
  2. Vertical Shift: There is no visible upward or downward shift of the curve.
  3. Reflections or Scaling: There is no reflection or noticeable vertical scaling; the shape remains the same.

Transformation:

To achieve the desired leftward shift, the transformation of the function y=xy = \sqrt{x} can be represented as: y=x+h,y = \sqrt{x + h}, where hh is the amount of horizontal shift.

From the graph:

  • The dotted curve starts at approximately x=4x = -4, compared to the original curve starting at x=0x = 0.
  • Therefore, h=4h = 4.

Final Transformed Function:

y=x+4.y = \sqrt{x + 4}.

Let me know if you'd like further clarification or a more detailed explanation!


Questions to Explore:

  1. How would the graph change if the function was shifted vertically instead?
  2. What would happen if we reflected the function over the x-axis?
  3. How can we identify the amount of shift just from the equation?
  4. Can you explain how compressions and stretches affect the graph of y=xy = \sqrt{x}?
  5. How do these transformations apply to other types of functions?

Tip:

Always start by identifying the transformations (horizontal shift, vertical shift, reflection, or scaling) by comparing key points of the original and transformed graphs.

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Math Problem Analysis

Mathematical Concepts

Function Transformations
Square Root Functions
Horizontal Shifts

Formulas

y = √(x + h)

Theorems

Transformation of Functions

Suitable Grade Level

Grades 8-10