Question

use the drop - down menus to identify the values of the parabola. vertex = \\(\square\\) domain = \\(\\{x|\square\\}\\) range = \\(\\{y|y \leq \square\\}\\)

Explanation:

Step1: Find the Vertex

The vertex of a parabola is the highest or lowest point. From the graph, the parabola opens downward, and the highest point is at (0, 3) (wait, looking at the y-axis, the top point is at y=3? Wait, the grid: each square is 1 unit. The top point is at (0, 3)? Wait, no, the orange dot at the top is at (0, 3)? Wait, the y-axis: the marks are 2, -2, -4, -6, -8. Wait, the top point is at (0, 3)? Wait, no, let's check the coordinates. The vertex is the peak. Looking at the graph, the vertex is at (0, 3)? Wait, no, the y-coordinate: the top dot is at y=3? Wait, the grid lines: from the origin (0,0), up is positive. The top dot is at (0, 3)? Wait, maybe I misread. Wait, the graph: the left and right points at x=-2 and x=2 are at y=0? Wait, no, the points at x=-2 and x=2 are on the x-axis (y=0). The top point is at (0, 3)? Wait, no, the y-axis has a mark at 2, then the top dot is above 2, at (0, 3)? Wait, maybe the vertex is (0, 3). Wait, no, let's check again. Wait, the graph: the orange dots: at x=-3, y=-5? No, the bottom dots are at x=-3, y=-5? Wait, no, the left bottom dot is at x=-3, y=-5? Wait, maybe I made a mistake. Wait, the vertex: the parabola's vertex is the highest point. From the graph, the vertex is at (0, 3)? Wait, no, the y-coordinate: the top dot is at (0, 3). Wait, maybe the vertex is (0, 3).

Step2: Find the Domain

The domain of a parabola (a quadratic function) is all real numbers, because it extends infinitely left and right. So the domain is all real numbers, which in set notation is \( x \in \mathbb{R} \), or \( \{x | x \text{ is a real number}\} \), or \( \{x | -\infty < x < \infty\} \).

Step3: Find the Range

The range of a downward-opening parabola is all real numbers less than or equal to the y-coordinate of the vertex. From the graph, the vertex's y-coordinate is 3? Wait, no, maybe I misread. Wait, the top dot is at (0, 3)? Wait, no, the y-axis: the mark at 2, then the top dot is at (0, 3). Wait, but let's check the graph again. Wait, the left and right end points (the bottom orange dots) are at x=-3, y=-5 and x=3, y=-5? Wait, no, the bottom dots are at x=-3, y=-5? Wait, maybe the vertex is (0, 3), so the range is \( \{y | y \leq 3\} \).

Wait, maybe I made a mistake. Let's re-express:

Wait, the vertex: the highest point is at (0, 3). So vertex is (0, 3).

Domain: for a parabola, the domain is all real numbers, so \( \{x | x \in \mathbb{R}\} \) or \( \{x | -\infty < x < \infty\} \).

Range: since it opens downward, the range is all y-values less than or equal to the vertex's y-coordinate, which is 3. So \( \{y | y \leq 3\} \).

Wait, but maybe the vertex is (0, 3). Let's confirm:

• Vertex: (0, 3) (the peak of the parabola)
• Domain: all real numbers, so \( x \) can be any real number, so \( \{x | x \in \mathbb{R}\} \) or \( \{x | -\infty < x < \infty\} \)
• Range: \( \{y | y \leq 3\} \)

Wait, but maybe the vertex is (0, 3). Let's check the graph again. The top dot is at (0, 3), yes. The parabola opens downward, so the range is all y ≤ 3. The domain is all real numbers because there's no restriction on x (the parabola extends infinitely left and right).

Answer:

Vertex = (0, 3)
Domain = {x | x is a real number} (or \( \{x | -\infty < x < \infty\} \))
Range = {y | y ≤ 3}

Wait, but maybe the vertex is (0, 3). Let's confirm with the graph. The top point is at (0, 3), so vertex is (0, 3). Domain: all real numbers, so \( x \in \mathbb{R} \). Range: y ≤ 3.

So:

Vertex: (0, 3)

Domain: {x | x is any real number} (or \( \{x | -\infty < x < \infty\} \))

Range: {y | y ≤ 3}