How do you find the coordinates of a vertex of a parabola using the quadratic equation?
To find the coordinates of the vertex of a parabola using the quadratic equation, you can follow these steps:
Step 1: Write the quadratic equation in standard form. The standard form of a quadratic equation is given by f(x) = ax^2 + bx + c, where a, b, and c are constants.
Step 2: Identify the values of a, b, and c from the quadratic equation.
Step 3: Use the formula for the x-coordinate of the vertex, which is given by x = -b / (2a). This formula is derived from the fact that the x-coordinate of the vertex lies at the axis of symmetry of the parabola, which is given by x = -b / (2a).
Step 4: Substitute the value of x into the quadratic equation to find the corresponding y-coordinate. Simply substitute the value of x into the original equation and evaluate to find the corresponding y-coordinate of the vertex.
Step 5: The coordinates of the vertex are given by (x, y), where x is the x-coordinate you found in step 3 and y is the y-coordinate you found in step 4.
It's important to note that if the quadratic equation is not in standard form, you may need to manipulate the equation to get it into the standard form before following these steps. Additionally, if the coefficient of x^2 (a) is positive, the vertex will be the lowest point on the parabola, whereas if a is negative, the vertex will be the highest point on the parabola.