Math Problem Statement

Use transformations to graph the given function a(x) = √(2x).

Solution

The problem involves graphing the function a(x)=2xa(x) = \sqrt{2x} using transformations.

Explanation:

  1. The base function is f(x)=xf(x) = \sqrt{x}, which is the square root function.
  2. The transformation involves multiplying xx by a factor of 2 inside the square root, resulting in a horizontal compression of the graph by a factor of 2\sqrt{2}. This means the graph becomes "narrower" as the xx-values are effectively scaled down.
  3. Domain: Since you cannot take the square root of a negative number, the domain of a(x)=2xa(x) = \sqrt{2x} is x0x \geq 0.

Observations:

  • The graph starts at the origin (0,0)(0, 0), as a(0)=20=0a(0) = \sqrt{2 \cdot 0} = 0.
  • It increases smoothly for positive xx-values.

Let me know if you'd like a step-by-step breakdown or need anything else explained.

Follow-up Questions:

  1. What are the differences between horizontal and vertical transformations?
  2. How does scaling inside the square root differ from scaling outside it?
  3. Can you determine the range of a(x)=2xa(x) = \sqrt{2x}?
  4. How would the graph look if the function were a(x)=2xa(x) = -\sqrt{2x}?
  5. How does the transformation affect the slope of the graph at different points?

Tip:

When working with transformations, always identify how the coefficient affects the xx-values (inside functions) or yy-values (outside functions).

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Math Problem Analysis

Mathematical Concepts

Graph transformations
Square root functions

Formulas

a(x) = √(2x)

Theorems

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Suitable Grade Level

Grades 9-11