Complete the square to re-write the quadratic function in vertex form:

Question

y, equals, minus, 9, x, squared, minus, 108, x, minus, 333
y=−9x
2
−108x−333

Answer 1

To rewrite the quadratic function \( y = -9x^2 - 108x - 333 \) in vertex form by completing the square, follow these steps:

1. Factor out the coefficient of \( x^2 \) from the first two terms: \[ y = -9(x^2 + 12x) - 333 \]

2. Complete the square inside the parentheses. To do this, take the coefficient of \( x \) (which is \( 12 \)), divide it by 2 to get \( 6 \), and then square it to obtain \( 36 \).

3. Add and subtract this square inside the parentheses: \[ y = -9(x^2 + 12x + 36 - 36) - 333 \] Simplifying this gives: \[ y = -9((x + 6)^2 - 36) - 333 \]

4. Distribute the -9: \[ y = -9(x + 6)^2 + 324 - 333 \] This simplifies to: \[ y = -9(x + 6)^2 - 9 \]

Now the quadratic function in vertex form is: \[ y = -9(x + 6)^2 - 9 \]

The vertex of the parabola represented by this equation is at \( (-6, -9) \).