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 \( y = 3x^2 - 12x + 16 \) in vertex form, we will complete the square.

1. First, factor out the coefficient of \( x^2 \) (which is 3) from the first two terms:

\[
y = 3(x^2 - 4x) + 16
\]

2. Next, complete the square for the expression \( x^2 - 4x \). To do this, take half of the coefficient of \( x \) (which is -4), square it, and add and subtract that value inside the parentheses. Half of -4 is -2, and squaring -2 gives us 4.

So, we rewrite the equation as:

\[
y = 3(x^2 - 4x + 4 - 4) + 16
\]

3. Then, rearranging the expression:

\[
y = 3((x - 2)^2 - 4) + 16
\]

4. Distributing the 3:

\[
y = 3(x - 2)^2 - 12 + 16
\]

5. Finally, combine like terms:

\[
y = 3(x - 2)^2 + 4
\]

Now, the quadratic function \( y = 3x^2 - 12x + 16 \) is rewritten in vertex form as:

\[
y = 3(x - 2)^2 + 4
\]

The vertex of the parabola is at the point \( (2, 4) \).