how do u solve for the vertex of a parabloa using the quadractic equation?
To solve for the vertex of a parabola using the quadratic equation, you can follow these steps:
Step 1: Write the quadratic equation in the form: y = ax^2 + bx + c
Step 2: Determine the values of a, b, and c from the given quadratic equation.
Step 3: Find the x-coordinate of the vertex by using the formula: x = -b / (2a). This formula is derived from the fact that the x-coordinate of the vertex lies on the axis of symmetry.
Step 4: Substitute the x-coordinate found in Step 3 back into the original equation to find the corresponding y-coordinate.
Step 5: The vertex of the parabola is given by the ordered pair (x, y), where x is the x-coordinate and y is the y-coordinate found.
Let's work through an example:
Example:
Given the quadratic equation: y = 2x^2 - 8x + 6, find the vertex.
Step 1: The quadratic equation is already in the form y = ax^2 + bx + c.
Step 2: From the given equation, we can identify that a = 2, b = -8, and c = 6.
Step 3: Use the formula x = -b / (2a) to find the x-coordinate of the vertex.
x = -(-8) / (2*2)
x = 8 / 4
x = 2
Step 4: Substitute x = 2 back into the original equation to find the y-coordinate.
y = 2(2)^2 - 8(2) + 6
y = 2(4) - 16 + 6
y = 8 - 16 + 6
y = -2
Step 5: The vertex is the ordered pair (x, y), where x = 2 and y = -2.
Vertex = (2, -2)
So, the vertex of the given parabola is (2, -2).