Question

Findtheminimumvalueoftheparabolay = x2 − 6x.


Simplify your answer and write it as a proper fraction, improper fraction, or integer.

Answer 1

To find the minimum value of the parabola given by the equation \( y = x^2 - 6x \), we can complete the square.

1. Start with the original equation:
\[
y = x^2 - 6x
\]

2. To complete the square, we need to take the coefficient of \( x \) (which is -6), divide it by 2, and square it:
\[
\left(-\frac{6}{2}\right)^2 = (-3)^2 = 9
\]

3. Add and subtract this square (9) inside the equation:
\[
y = (x^2 - 6x + 9) - 9
\]

4. Now, rewrite the equation:
\[
y = (x - 3)^2 - 9
\]

Now we have the equation in vertex form \( y = (x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. Here, the vertex is at \((3, -9)\).

Therefore, the minimum value of the parabola (since it opens upwards) occurs at the vertex \( y = -9 \).

Thus, the minimum value of the parabola \( y = x^2 - 6x \) is:
\[
\boxed{-9}
\]