By Amy Harris • Updated Aug 30, 2022
Converting a quadratic equation to vertex form can be a precise task that benefits from a solid grasp of algebraic techniques. The vertex form—y = a(x – h)^2 + k—encapsulates the parabola’s key feature: its vertex, located at (h, k). In this tutorial, we’ll walk through each step to transform a standard quadratic into this elegant representation.
Start with the equation in standard form: y = ax^2 + bx + c. For example, y = 2x^2 + 8x – 10 is already in standard form, whereas y – 8x = 2x^2 – 10 is not; adding 8x to both sides yields the correct format.
Move the constant term to the left side by adding or subtracting it. In y = 2x^2 + 8x – 10, the constant is –10; add 10 to both sides: y + 10 = 2x^2 + 8x.
Factor out the coefficient of the squared term, a. Here, a = 2, giving: y + 10 = 2(x^2 + 4x).
Complete the square inside the parentheses. Divide the coefficient of the linear term by 2 (4 ÷ 2 = 2), square the result (2^2 = 4), and insert it: y + 10 = 2(x^2 + 4x + 4).
Adjust the constant on the left side. Multiply a by the square added in Step 4: 2 × 4 = 8. Add this to the existing constant: y + 18 = 2(x^2 + 4x + 4).
The expression inside the parentheses is now a perfect square: (x + 2)^2. Rewrite the equation: y + 18 = 2(x + 2)^2.
Isolate y by moving the constant back to the right side: subtract 18 from both sides. The final vertex form is y = 2(x + 2)^2 – 18. Here, h = –2 and k = –18, so the vertex is (–2, –18).
