Question
Find the minimum value of the parabola y = x^2 − 5.
Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Answers
GPT-4o mini
The given equation of the parabola is
\[
y = x^2 - 5.
\]
This is a standard form of a parabola opening upwards, where the vertex of the parabola is at the minimum point. The general form of a parabola is
\[
y = ax^2 + bx + c,
\]
where \( a > 0 \) indicates that it opens upwards.
To find the vertex \( (h, k) \), we use the formula for the x-coordinate of the vertex:
\[
h = -\frac{b}{2a}.
\]
In our equation, we have:
- \( a = 1 \)
- \( b = 0 \)
- \( c = -5 \)
Calculating \( h \):
\[
h = -\frac{0}{2 \cdot 1} = 0.
\]
Next, we substitute \( x = 0 \) back into the equation to find \( k \):
\[
k = (0)^2 - 5 = -5.
\]
Thus, the vertex of the parabola is at the point \( (0, -5) \).
Since this is where the minimum occurs, the minimum value of \( y \) is
\[
\boxed{-5}.
\]
\[
y = x^2 - 5.
\]
This is a standard form of a parabola opening upwards, where the vertex of the parabola is at the minimum point. The general form of a parabola is
\[
y = ax^2 + bx + c,
\]
where \( a > 0 \) indicates that it opens upwards.
To find the vertex \( (h, k) \), we use the formula for the x-coordinate of the vertex:
\[
h = -\frac{b}{2a}.
\]
In our equation, we have:
- \( a = 1 \)
- \( b = 0 \)
- \( c = -5 \)
Calculating \( h \):
\[
h = -\frac{0}{2 \cdot 1} = 0.
\]
Next, we substitute \( x = 0 \) back into the equation to find \( k \):
\[
k = (0)^2 - 5 = -5.
\]
Thus, the vertex of the parabola is at the point \( (0, -5) \).
Since this is where the minimum occurs, the minimum value of \( y \) is
\[
\boxed{-5}.
\]