Convert standard form to vertex form is an effective tool for graphing quadratic functions. Quadratic equation can be expressed in two forms. They are standard form and vertex form. The standard form of quadratic equation is ax2 + b x + c = 0. The vertex form is y= a (x - h)2 + k where (h, k) is the vertex. We can convert quadratic function from standard form to vertex form by completing the square. A quadratic function is much easier to graph when written in vertex form.
Example 1: Rewrite the equation y = 2x2 - 8x + 1 in the form y = a(x - h)2 + k by completing the square.
Solution: For solving the equation y = 2x2 - 8x + 1
=> First, factor out a 2. This is done because in order to complete the square,
=> The coefficient of x2 must be 1.
=> y = 2x2 – 8x + 1
=> y = 2(x2 – 4x) + 1
=> Now, complete the square. Take half of the coefficient of x, 4/ 2 = 2 and square it
=> 22 = 4.
=> Add this to the equation. Since it is added, it must also be subtracted to keep the equation equal. Then, simplify.
=> y = 2(x2 – 4x + 4 – 4) + 1
=> y = 2(x2 – 4x + 4) + (-8 + 1)
=> y = 2(x-2)2 - 7
Example 2: Rewrite the equation y = 4x2 - 24x + 46 in the form y = a(x - h) 2 + k by completing the square
Solution: x2 needs a coefficient of 1 in order to complete the square.
=> y = 4x2 - 24x + 46
=> y = 4(x2 - 6x) + 46
=> y = 4(x2 - 6x + 9 – 9) + 46
=> y = 4(x2 - 6x + 9)+ (-36 + 46)
=> y = 4(x – 3)2 + 10
=> This equation is in vertex form is y = 4(x – 3)2 + 10