Math Problem Statement

y=x2+3

Solution

The equation you've provided is:

y=x2+3y = x^2 + 3

This represents a quadratic function. Let's break it down:

  1. Parabola: The graph of this equation is a parabola, opening upwards, since the coefficient of x2x^2 is positive.
  2. Vertex: The vertex of the parabola is at (0,3)(0, 3), because the constant term +3+3 shifts the graph upwards by 3 units.
  3. Axis of symmetry: The axis of symmetry is the vertical line x=0x = 0, which passes through the vertex.
  4. Y-intercept: The y-intercept is at (0,3)(0, 3), because when x=0x = 0, y=3y = 3.
  5. X-intercepts: There are no real x-intercepts, as the equation x2+3=0x^2 + 3 = 0 does not have real solutions (it only has complex solutions).

Would you like more details or have any questions?

Here are five questions related to this:

  1. How would the graph change if we modified the equation to y=x23y = x^2 - 3?
  2. What are the complex solutions to the equation x2+3=0x^2 + 3 = 0?
  3. How does the value of the constant term affect the graph of a quadratic equation?
  4. Can you describe how to find the axis of symmetry for any quadratic function?
  5. What is the general form of a quadratic equation?

Tip: For quadratic equations, the vertex form is useful for identifying the vertex and transformations of the graph easily!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Parabolas

Formulas

Standard form of a quadratic equation: y = ax^2 + bx + c
Axis of symmetry: x = -b/2a

Theorems

No real solutions theorem for x^2 + c = 0 when c > 0
Vertex form transformation theorem

Suitable Grade Level

Grades 9-10