Question

Findtheminimumvalueoftheparabolay = x2 + 9.


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

Answer 1

The given equation of the parabola is \( y = x^2 + 9 \).

This is a standard form of a quadratic equation \( y = ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = 0 \)
- \( c = 9 \)

Since \( a > 0 \), the parabola opens upward. The minimum value of the parabola occurs at its vertex.

The x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting \( a \) and \( b \):
\[
x = -\frac{0}{2 \cdot 1} = 0
\]

Now, substitute \( x = 0 \) back into the equation to find the corresponding \( y \)-value:
\[
y = (0)^2 + 9 = 9
\]

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