Math Problem Statement

Identify and explain the properties of the function y = x^2.

Solution

The image shows the function y=x2y = x^2, which represents a basic quadratic function. Here’s a breakdown of key properties for y=x2y = x^2:

  1. Shape: The graph of y=x2y = x^2 is a parabola that opens upwards.
  2. Vertex: The vertex of this parabola is at the origin, (0,0)(0, 0).
  3. Axis of Symmetry: The axis of symmetry is the vertical line x=0x = 0.
  4. Domain: The domain of y=x2y = x^2 is all real numbers (,)(-\infty, \infty).
  5. Range: Since y=x2y = x^2 only produces non-negative values, its range is [0,)[0, \infty).

Would you like to see a graph of this function, or perhaps explore transformations of this parabola?

Here are some related questions for further exploration:

  1. What happens to the graph of y=x2y = x^2 if you add or subtract a constant?
  2. How does the graph change if we replace x2x^2 with (xh)2(x - h)^2 for some constant hh?
  3. What if we introduce a coefficient, such as y=ax2y = ax^2? How does the value of aa affect the graph?
  4. How do we find the roots of a quadratic function like y=x2cy = x^2 - c?
  5. What is the relationship between the derivative of y=x2y = x^2 and the slope of the tangent to the curve at any given point?

Tip: For quadratic functions, remember that the vertex form y=a(xh)2+ky = a(x - h)^2 + k can be particularly useful for identifying the vertex and transformations of the basic parabola.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Parabolas

Formulas

y = x^2

Theorems

Vertex Form
Symmetry of Parabolas

Suitable Grade Level

Grades 8-10