Answer:
1. Coordinates of the vertex of the graph: (-3, -1)
2. The graph opens up.
2. The graph has a minimum.
Explanation:
Currently, f(x) = (x + 3)^2 - 1 is in the vertex form of a quadratic equation. The general equation of the vertex form is given by:
f(x) = a(x - h)^2 + k, where
- a is a constant determining whether the parabola opens up or down (this also means it determines whether the vertex is a maximum or minimum),
- and (h, k) are the coordinates of the vertex.
1. When dealing with the vertex form, the h coordinate becomes the opposite of what it is inside the parentheses. Thus, h is -3 and k is -1. Therefore, the coordinates of the vertex are (-3, -1).
2. Question #1: We can imagine that there's an imaginary 1 in front of (x + 3)^2 and thus a = 1.
- When a > 0, the parabola opens up.
- When a < 0, the parabola opens down.
Since a = 1 and a > 0, the parabola opens up.
2. Question #2: All parabolas that open up will by definition have a minimum. Thus, the graph of the function f has a minimum.