Question
Find the minimum value of the parabola: y = x^2 - 4. 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 given by the equation \( y = x^2 - 4 \), we first note that this parabola opens upwards because the coefficient of \( x^2 \) (which is 1) is positive.
A parabola in the standard form \( y = ax^2 + bx + c \) reaches its minimum value at the vertex given by the formula:
\[
x = -\frac{b}{2a}
\]
For our equation, we have:
- \( a = 1 \)
- \( b = 0 \)
- \( c = -4 \)
Now, we can substitute the values into the vertex formula:
\[
x = -\frac{0}{2 \cdot 1} = 0
\]
Next, we substitute \( x = 0 \) back into the equation \( y = x^2 - 4 \) to find the corresponding \( y \)-value:
\[
y = 0^2 - 4 = -4
\]
Thus, the minimum value of the parabola \( y = x^2 - 4 \) is:
\[
\boxed{-4}
\]
A parabola in the standard form \( y = ax^2 + bx + c \) reaches its minimum value at the vertex given by the formula:
\[
x = -\frac{b}{2a}
\]
For our equation, we have:
- \( a = 1 \)
- \( b = 0 \)
- \( c = -4 \)
Now, we can substitute the values into the vertex formula:
\[
x = -\frac{0}{2 \cdot 1} = 0
\]
Next, we substitute \( x = 0 \) back into the equation \( y = x^2 - 4 \) to find the corresponding \( y \)-value:
\[
y = 0^2 - 4 = -4
\]
Thus, the minimum value of the parabola \( y = x^2 - 4 \) is:
\[
\boxed{-4}
\]