Question

Find the minimum value of the parabola: y = x^2 + x + 7. 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 \( y = x^2 + x + 7 \), we can use the formula for the vertex of a parabola in standard form \( y = ax^2 + bx + c \).

1. Identify the coefficients:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 7 \)

2. The x-coordinate of the vertex of a parabola is given by the formula:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{1}{2 \cdot 1} = -\frac{1}{2}
\]

3. Now, substitute \( x = -\frac{1}{2} \) back into the equation to find the corresponding y-coordinate (the minimum value of \( y \)):
\[
y = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + 7
\]
Calculate each term:
\[
\left(-\frac{1}{2}\right)^2 = \frac{1}{4}
\]
\[
-\frac{1}{2} = -\frac{1}{2}
\]
Adding these:
\[
y = \frac{1}{4} - \frac{1}{2} + 7
\]
Convert \(-\frac{1}{2}\) to a fraction with a denominator of 4:
\[
-\frac{1}{2} = -\frac{2}{4}
\]
So we have:
\[
y = \frac{1}{4} - \frac{2}{4} + 7 = \frac{1 - 2}{4} + 7 = -\frac{1}{4} + 7
\]
Convert 7 to a fraction:
\[
7 = \frac{28}{4}
\]
Now add the fractions:
\[
y = -\frac{1}{4} + \frac{28}{4} = \frac{27}{4}
\]

Thus, the minimum value of the parabola is:
\[
\boxed{\frac{27}{4}}
\]