Several statements are provided. Select the statement that is true for the equation y=−(x+2)2−6

Question

.

Statement #1: The parabola opens down, has an axis of symmetry at x=−2
, and a vertex at (−2,−6)
.

Statement #2: The parabola opens down, has an axis of symmetry at x=2
, and a vertex at (2, 6)
.

Statement #3: The parabola opens up, has an axis of symmetry at x=−2
, and a vertex at (−2,6)
.

Statement #4: The parabola opens down, has an axis of symmetry at x=2
, and a vertex at (−2,−6)
.

Answer 1

To analyze the given equation \( y = - (x + 2)^2 - 6 \), we can derive the characteristics of the parabola.

1. The equation is in the vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex and \( a \) determines the direction the parabola opens.
2. Here, \( a = -1 \) (negative value), indicating that the parabola opens downward.
3. The vertex can be identified from the equation as \( (h, k) \). In this case, \( h = -2 \) and \( k = -6 \), making the vertex \( (-2, -6) \).
4. The axis of symmetry is at the line \( x = h \), which in this case is \( x = -2 \).

Now let's evaluate the statements:

• Statement #1: The parabola opens down, has an axis of symmetry at \( x = -2 \), and a vertex at \( (-2, -6) \). (True)

• Statement #2: The parabola opens down, has an axis of symmetry at \( x = 2 \), and a vertex at \( (2, 6) \). (False)

• Statement #3: The parabola opens up, has an axis of symmetry at \( x = -2 \), and a vertex at \( (-2, 6) \). (False)

• Statement #4: The parabola opens down, has an axis of symmetry at \( x = 2 \), and a vertex at \( (-2, -6) \). (False)

The true statement is Statement #1.