Question
Several statements are provided. Select the statement that is true for the equation y=−(x+2)2−6
.
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)
.
.
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)
.
Answers
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Answered by
GPT-4o mini
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**.
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**.
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