Question

Findtheminimumvalueoftheparabola
y
=
x
2

+
x
+
11
4
.


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

\[
y = x^2 + x + \frac{11}{4},
\]

we can use the formula for the vertex of a parabola, which occurs at \( x = -\frac{b}{2a} \), where \( a \) is the coefficient of \( x^2 \), and \( b \) is the coefficient of \( x \).

Here, \( a = 1 \) and \( b = 1 \).

Calculating the x-coordinate of the vertex:

\[
x = -\frac{b}{2a} = -\frac{1}{2 \times 1} = -\frac{1}{2}.
\]

Next, we substitute \( x = -\frac{1}{2} \) back into the equation to find the minimum value of \( y \):

\[
y = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + \frac{11}{4}.
\]

Calculating each term:

\[
\left(-\frac{1}{2}\right)^2 = \frac{1}{4},
\]
\[
-\frac{1}{2} = -\frac{2}{4},
\]
\[
\frac{11}{4} = \frac{11}{4}.
\]

Now, combine these values:

\[
y = \frac{1}{4} - \frac{2}{4} + \frac{11}{4} = \frac{1 - 2 + 11}{4} = \frac{10}{4} = \frac{5}{2}.
\]

Thus, the minimum value of the given parabola is

\[
\boxed{\frac{5}{2}}.
\]