y=x^2+3

The equation \( y = x^2 + 3 \) represents a quadratic function. Here are some key features of this function:

1. Shape: The graph of this function is a parabola that opens upwards.

2. Vertex: The vertex of the parabola can be found using the formula for the vertex of a quadratic equation in the form \( y = ax^2 + bx + c \). In this case, \( a = 1 \), \( b = 0 \), and \( c = 3 \). The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} = -\frac{0}{2 \cdot 1} = 0 \). Substituting \( x = 0 \) back into the equation gives \( y = 0^2 + 3 = 3 \). Thus, the vertex is at the point \( (0, 3) \).

3. Axis of symmetry: The axis of symmetry is the vertical line that passes through the vertex, which is \( x = 0 \).

4. Y-intercept: The y-intercept occurs when \( x = 0 \). As calculated, \( y = 3 \), so the y-intercept is \( (0, 3) \).

5. X-intercepts: To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 + 3 \] This gives \( x^2 = -3 \), which means there are no real x-intercepts (the parabola does not cross the x-axis).

6. Domain and Range:

   • The domain of the function is all real numbers: \( (-\infty, \infty) \).
   • The range starts from the vertex's y-value and goes up to infinity: \( [3, \infty) \).

If you have specific questions regarding this function or need assistance with related concepts, feel free to ask!