Question
Find the minimum value of the parabola: y = x^2 - 4x. Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Answers
GPT-4o mini
To find the minimum value of the parabola defined by the equation \( y = x^2 - 4x \), we can first complete the square.
1. Start with the quadratic expression:
\[
y = x^2 - 4x
\]
2. To complete the square, we take the coefficient of \( x \), which is \(-4\), halve it to get \(-2\), and then square it to get \(4\):
\[
y = (x^2 - 4x + 4) - 4
\]
3. This can be rewritten as:
\[
y = (x - 2)^2 - 4
\]
4. The vertex form of a parabola \( y = a(x-h)^2 + k \) indicates that the vertex \((h, k)\) is the minimum point when \( a > 0 \). In this case, \( a = 1 \), which is positive, and the vertex is located at:
\[
(h, k) = (2, -4)
\]
5. Therefore, the minimum value of the parabola is:
\[
-4
\]
This is confirmed since the parabola opens upwards and the vertex represents the lowest point on the graph.
Thus, the minimum value of the parabola is:
\[
\boxed{-4}
\]
1. Start with the quadratic expression:
\[
y = x^2 - 4x
\]
2. To complete the square, we take the coefficient of \( x \), which is \(-4\), halve it to get \(-2\), and then square it to get \(4\):
\[
y = (x^2 - 4x + 4) - 4
\]
3. This can be rewritten as:
\[
y = (x - 2)^2 - 4
\]
4. The vertex form of a parabola \( y = a(x-h)^2 + k \) indicates that the vertex \((h, k)\) is the minimum point when \( a > 0 \). In this case, \( a = 1 \), which is positive, and the vertex is located at:
\[
(h, k) = (2, -4)
\]
5. Therefore, the minimum value of the parabola is:
\[
-4
\]
This is confirmed since the parabola opens upwards and the vertex represents the lowest point on the graph.
Thus, the minimum value of the parabola is:
\[
\boxed{-4}
\]