Question
Find the vertex of the parabola: y = x^2 + 7x + 6. Simplify both coordinates and write them as proper fractions, improper fractions, or integers.
Answers
GPT-4o mini
To find the vertex of the parabola given by the equation \( y = x^2 + 7x + 6 \), we can start by using the vertex formula.
The x-coordinate of the vertex for a parabola in the form \( y = ax^2 + bx + c \) is given by:
\[
x = -\frac{b}{2a}
\]
In this case, \( a = 1 \) and \( b = 7 \). Plugging in these values:
\[
x = -\frac{7}{2 \cdot 1} = -\frac{7}{2}
\]
Next, we substitute this x-value back into the original equation to find the y-coordinate of the vertex:
\[
y = \left(-\frac{7}{2}\right)^2 + 7\left(-\frac{7}{2}\right) + 6
\]
Calculating each term:
1. \( \left(-\frac{7}{2}\right)^2 = \frac{49}{4} \)
2. \( 7 \left(-\frac{7}{2}\right) = -\frac{49}{2} \), and converting this to a fraction with a denominator of 4 gives us \( -\frac{49}{2} = -\frac{98}{4} \)
3. The constant term \( 6 = \frac{24}{4} \)
Now we can combine these terms:
\[
y = \frac{49}{4} - \frac{98}{4} + \frac{24}{4}
\]
Putting it all together, we get:
\[
y = \frac{49 - 98 + 24}{4} = \frac{-25}{4}
\]
Thus, the coordinates of the vertex are:
\[
\left(-\frac{7}{2}, -\frac{25}{4}\right)
\]
To summarize, the vertex of the parabola \( y = x^2 + 7x + 6 \) is
\[
\boxed{\left(-\frac{7}{2}, -\frac{25}{4}\right)}
\]
The x-coordinate of the vertex for a parabola in the form \( y = ax^2 + bx + c \) is given by:
\[
x = -\frac{b}{2a}
\]
In this case, \( a = 1 \) and \( b = 7 \). Plugging in these values:
\[
x = -\frac{7}{2 \cdot 1} = -\frac{7}{2}
\]
Next, we substitute this x-value back into the original equation to find the y-coordinate of the vertex:
\[
y = \left(-\frac{7}{2}\right)^2 + 7\left(-\frac{7}{2}\right) + 6
\]
Calculating each term:
1. \( \left(-\frac{7}{2}\right)^2 = \frac{49}{4} \)
2. \( 7 \left(-\frac{7}{2}\right) = -\frac{49}{2} \), and converting this to a fraction with a denominator of 4 gives us \( -\frac{49}{2} = -\frac{98}{4} \)
3. The constant term \( 6 = \frac{24}{4} \)
Now we can combine these terms:
\[
y = \frac{49}{4} - \frac{98}{4} + \frac{24}{4}
\]
Putting it all together, we get:
\[
y = \frac{49 - 98 + 24}{4} = \frac{-25}{4}
\]
Thus, the coordinates of the vertex are:
\[
\left(-\frac{7}{2}, -\frac{25}{4}\right)
\]
To summarize, the vertex of the parabola \( y = x^2 + 7x + 6 \) is
\[
\boxed{\left(-\frac{7}{2}, -\frac{25}{4}\right)}
\]