Question

Transform the quadratic equation from standard form to vertex form and state the vertex point.

y
=
x
2
+
10
x
+
12

(1 point)
Responses

y
=
(
x
+
5
)
2
−
37
and
(
−
5
,
−
37
)
y equals left parenthesis x plus 5 right parenthesis squared minus 37 and left parenthesis negative 5 comma negative 37 right parenthesis

y
=
(
x
+
5
)
2
−
13
and
(
−
5
,
−
13
)
y equals left parenthesis x plus 5 right parenthesis squared minus 13 and left parenthesis negative 5 comma negative 13 right parenthesis

y
=
(
x
+
5
)
2
−
13
and
(
5
,
−
13
)
y equals left parenthesis x plus 5 right parenthesis squared minus 13 and left parenthesis 5 comma negative 13 right parenthesis

y
=
(
x
+
5
)
2
+
13
and
(
−
5
,
−
13
)

Answer 1

To transform the quadratic equation \( y = x^2 + 10x + 12 \) from standard form to vertex form, we need to complete the square.

1. Start with the equation:
\[
y = x^2 + 10x + 12
\]

2. Focus on the quadratic and linear terms (\(x^2 + 10x\)):
- To complete the square, take half of the coefficient of \(x\) (which is \(10\)), square it, and then add and subtract that square inside the equation.
- Half of \(10\) is \(5\), and squaring it gives \(25\).

3. Rewrite the equation:
\[
y = (x^2 + 10x + 25) - 25 + 12
\]
\[
y = (x + 5)^2 - 13
\]

4. Now, we have the equation in vertex form, which is:
\[
y = (x + 5)^2 - 13
\]

5. The vertex of the parabola is given by the point \((h, k)\) in the vertex form equation \(y = a(x - h)^2 + k\). In this case, \(h = -5\) and \(k = -13\).

So the vertex is \((-5, -13)\).

Thus, the answer is:
\[
y = (x + 5)^2 - 13 \quad \text{and} \quad (-5, -13)
\]