Question
Find the vertex of the parabola: y = x^2 + 2x + 7/4. Simplify both coordinates and write them as proper fractions, improper fractions, or integers. Should I enter x = (answer) or just (answer)
Answers
GPT-4o mini
To find the vertex of the parabola given by the equation \( y = x^2 + 2x + \frac{7}{4} \), we can use the formula for the x-coordinate of the vertex of a parabola in the form \( y = ax^2 + bx + c \), which is given by:
\[
x = -\frac{b}{2a}
\]
In our case, \( a = 1 \) and \( b = 2 \). Plugging in these values:
\[
x = -\frac{2}{2(1)} = -\frac{2}{2} = -1
\]
Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting \( x = -1 \) back into the equation:
\[
y = (-1)^2 + 2(-1) + \frac{7}{4}
\]
\[
= 1 - 2 + \frac{7}{4}
\]
\[
= -1 + \frac{7}{4}
\]
To combine \(-1\) and \(\frac{7}{4}\), we convert \(-1\) to a fraction with a denominator of 4:
\[
-1 = -\frac{4}{4}
\]
Now we can combine:
\[
-\frac{4}{4} + \frac{7}{4} = \frac{3}{4}
\]
Thus, the vertex of the parabola is \((-1, \frac{3}{4})\).
In conclusion, the coordinates of the vertex are:
\[
\boxed{(-1, \frac{3}{4})}
\]
When entering your answer, if asked only for \( x \), you should enter \( -1 \). If asked for the complete vertex, you would enter \( (-1, \frac{3}{4}) \).
\[
x = -\frac{b}{2a}
\]
In our case, \( a = 1 \) and \( b = 2 \). Plugging in these values:
\[
x = -\frac{2}{2(1)} = -\frac{2}{2} = -1
\]
Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting \( x = -1 \) back into the equation:
\[
y = (-1)^2 + 2(-1) + \frac{7}{4}
\]
\[
= 1 - 2 + \frac{7}{4}
\]
\[
= -1 + \frac{7}{4}
\]
To combine \(-1\) and \(\frac{7}{4}\), we convert \(-1\) to a fraction with a denominator of 4:
\[
-1 = -\frac{4}{4}
\]
Now we can combine:
\[
-\frac{4}{4} + \frac{7}{4} = \frac{3}{4}
\]
Thus, the vertex of the parabola is \((-1, \frac{3}{4})\).
In conclusion, the coordinates of the vertex are:
\[
\boxed{(-1, \frac{3}{4})}
\]
When entering your answer, if asked only for \( x \), you should enter \( -1 \). If asked for the complete vertex, you would enter \( (-1, \frac{3}{4}) \).