Math Problem Statement

what is y=x²

Solution

The equation y=x2y = x^2 represents a parabola in the coordinate plane.

Key Features:

  1. Shape: It is a U-shaped curve that opens upwards.
  2. Vertex: The lowest point of the parabola is at the origin, (0,0)(0, 0).
  3. Axis of Symmetry: The line x=0x = 0 (the y-axis) is the axis of symmetry.
  4. Domain: The domain is all real numbers, (,)(-\infty, \infty).
  5. Range: The range is [0,)[0, \infty), as yy is always non-negative.
  6. Behavior:
    • For x>0x > 0, yy increases as xx increases.
    • For x<0x < 0, yy increases as x|x| increases.

Explanation of the Equation:

  • The x2x^2 term means that the value of yy depends on the square of xx. Squaring any real number always results in a non-negative value.

Would you like a plot of this function? Or further exploration?

Related Questions:

  1. How do you find the vertex of a parabola in general form?
  2. How do transformations (e.g., y=(xh)2+ky = (x - h)^2 + k) change the graph of a parabola?
  3. How do you determine the focus and directrix of y=x2y = x^2?
  4. How does the graph change when you add a coefficient to x2x^2, like y=ax2y = ax^2?
  5. What are the real-world applications of parabolic curves?

Tip:

To better understand the parabola, analyze its symmetry and how changes to xx affect yy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Quadratic Equations
Graphing

Formulas

y = x^2

Theorems

Properties of parabolas
Symmetry in quadratic functions

Suitable Grade Level

Grades 7-9