Question

Rico is asked to graph a parabola given the equation y=−x2−2x . At which point would he graph the vertex of the parabola?(1 point)
Responses

(−1,1)
left parenthesis negative 1 comma 1 right parenthesis

(1,2)
left parenthesis 1 comma 2 right parenthesis

(−1,2)
left parenthesis negative 1 comma 2 right parenthesis

(−1,0)

Answer 1

To find the vertex of the parabola given by the equation \( y = -x^2 - 2x \), we can rewrite the equation in vertex form.

1. Start with the given equation:
\[
y = -x^2 - 2x
\]

2. Factor out the negative sign from the quadratic expression:
\[
y = -(x^2 + 2x)
\]

3. Complete the square for the expression inside the parentheses:
- Take half of the coefficient of \( x \) (which is 2), square it, and add it inside the parentheses. Half of 2 is 1, and squaring it gives us 1.
- Thus we can rewrite the equation as:
\[
y = -\left[(x^2 + 2x + 1) - 1\right]
\]
\[
y = -\left[(x + 1)^2 - 1\right]
\]
\[
y = -(x + 1)^2 + 1
\]

4. Now we can identify the vertex from the completed square form \( y = -1(x + 1)^2 + 1 \). The vertex of the parabola in the form \( y = a(x - h)^2 + k \) is given by the point \( (h, k) \).

Here, \( h = -1 \) and \( k = 1 \), so the vertex is at the point:
\[
(-1, 1)
\]

Thus, Rico would graph the vertex at the point \( (-1, 1) \).

The correct response is:
\[
\text{(−1, 1)}
\]