Question

Complete the square to re-write the quadratic function in vertex form: y, equals, 3, x, squared, minus, 12, x, plus, 16 y=3x 2 −12x 16

Answer 1

To rewrite the quadratic function y = 3x² - 12x + 16 in vertex form, factor out the leading coefficient, complete the square for the x terms, and simplify to get y = 3(x - 2)² + 4, where the vertex of the parabola is (2, 4).

Step-by-step explanation:

To complete the square and rewrite the quadratic function y = 3x² - 12x + 16 in vertex form, follow these steps:

1. First, factor out the coefficient of the x² term (which is 3) from the x terms of the quadratic. This gives us y = 3(x² - 4x) + 16.
2. Next, to complete the square inside the parentheses, take half of the x-coefficient (-4), square it, and add it inside the parentheses: (-4/2)² = 4. Then, add and subtract this value inside the parentheses y = 3(x² - 4x + 4 - 4) + 16.
3. Combine the terms inside the parentheses to form a perfect square, and adjust the constant outside: y = 3((x - 2)² - 4) + 16.
4. Finally, distribute the 3, combine the constant terms, and write the complete vertex form: y = 3(x - 2)² + 4.

The vertex form of the function is now y = 3(x - 2)² + 4, where (2, 4) is the vertex of the parabola.