Answer:
Explanation:
To solve the equation (x + 3)² = 4(y + 2) and determine the characteristics of the graph, we can use the standard form of a parabolic equation: (x - h)² = 4p(y - k), where (h, k) represents the vertex and p represents the distance between the vertex and the focus.
1. Comparing the given equation to the standard form, we can see that h = -3 and k = -2. Therefore, the vertex of the parabola is (-3, -2).
2. To find the value of p, we can rewrite the equation as (x + 3)² = 4(y + 2) and compare it to the standard form. We can see that 4p = 4, so p = 1.
3. Since p = 1, the distance between the vertex and the focus is 1 unit. Thus, the focus of the parabola is (-3, -1).
4. The axis of symmetry is a vertical line that passes through the vertex of the parabola. In this case, the equation of the axis of symmetry is x = -3.
5. To determine the concavity of the parabola, we can examine the coefficient of y in the equation. Since it is positive, the parabola opens upward and is concave up.
6. The parabola has a line of reflection (LR) that is parallel to the axis of symmetry. In this case, the LR is the line x = -3.
7. The endpoints of the LR correspond to the y-values that create a horizontal line with the same slope as the parabola. In this case, the endpoints of the LR are y = -2.
Concavity: The parabola opens upward and is concave up.
Vertex: (-3, -2)
Focus: (-3, -1)
Axis of Symmetry: x = -3
Direction: The parabola opens upward.
LR: x = -3
Endpoints of LR: y = -2