What is how to find vertex of parabola?

The vertex of a parabola is the point where the parabola changes direction. It's either the minimum point (for parabolas that open upwards) or the maximum point (for parabolas that open downwards). Here's how to find it:

1. Standard Form:

• If the parabola is given in standard form: y = ax² + bx + c, the x-coordinate of the vertex is found using the formula:

  • h = -b / 2a

  • To find the y-coordinate (k), substitute the value of 'h' back into the equation: k = a(h)² + b(h) + c

  • Therefore, the vertex is at the point (h, k).

2. Vertex Form:

• If the parabola is given in vertex form: y = a(x - h)² + k, then the vertex is simply (h, k). Note that the 'h' value in this form has the opposite sign of what appears in the equation.

3. Finding the Vertex from Intercepts (If Applicable):

• If you know the x-intercepts (roots) of the parabola, say x₁ and x₂, the x-coordinate of the vertex is the average of the intercepts:

  • h = (x₁ + x₂) / 2

  • Then, find the y-coordinate (k) by substituting 'h' back into the original equation.

Important Concepts:

• Axis of Symmetry: The vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = h, where 'h' is the x-coordinate of the vertex.

• Minimum and Maximum Values: The y-coordinate of the vertex represents the minimum value of the parabola if 'a' (the coefficient of x²) is positive, and the maximum value if 'a' is negative.

• Quadratic Formula: While not directly used to find the vertex (unless using intercepts method described above), understanding the quadratic formula is crucial for finding the roots (x-intercepts) of the parabola, which can then be used to find the vertex.