Question

six graphs are shown below. click on the tabs to see them.
graph 1 graph 2 graph 3 graph 4 graph 5 graph 6
(graph 3 shows a coordinate plane with a triangle; axes labeled x, y with range -2 to 2)
check the box corresponding to each graph that represents a function.
☑️ graph 1 ☐ graph 2 ☐ graph 3 ☑️ graph 4 ☐ graph 5 ☐ graph 6 ☐ none of the grap

Explanation:

Step1: Recall Vertical Line Test

A graph represents a function if no vertical line intersects the graph more than once.

Step2: Apply to Graph 3

For Graph 3, draw a vertical line (e.g., \(x = 0\), \(x = 0.5\), etc.). At \(x = 0\), the vertical line intersects the graph at two points (the top horizontal segment and the left/right sides? Wait, no, looking at the triangle: the vertical line through the y - axis (x = 0) intersects the triangle at the top vertex (y = 1) and the bottom vertex (y=- 1)? Wait, no, the triangle has a base at y = 1 (from x=-1 to x = 1) and a vertex at (0, - 1). Wait, no, when we apply the vertical line test: take any x - value. For x between - 1 and 1 (excluding the endpoints?), wait, the triangle: left side from (-1,1) to (0, - 1), right side from (1,1) to (0, - 1), and top side from (-1,1) to (1,1). Wait, if we take a vertical line at x = 0.5 (between 0 and 1), it intersects the top side (y = 1) and the right side? No, the right side is from (1,1) to (0, - 1). So at x = 0.5, the vertical line will intersect the top side (y = 1) and the right side? Wait, no, the top side is horizontal from x=-1 to x = 1 at y = 1. The right side is from (1,1) to (0, - 1). So for x in (-1,1), the vertical line will intersect the top horizontal line (y = 1) and the slant side? Wait, no, actually, the triangle is a closed figure. Wait, no, the graph is a triangle with vertices at (-1,1), (1,1), and (0, - 1). So when we apply the vertical line test: for any x in (-1,1), a vertical line at that x will intersect the top horizontal segment (y = 1) and the slant side (either left or right) at one point? Wait, no, let's take x = 0: vertical line x = 0 intersects the top segment at (0,1) and the bottom vertex at (0, - 1)? Wait, no, the bottom vertex is (0, - 1), and the top segment is from (-1,1) to (1,1), so at x = 0, the vertical line intersects the top segment at (0,1) and the bottom vertex at (0, - 1)? Wait, no, the bottom vertex is a single point. Wait, maybe I made a mistake. Let's re - examine: the triangle has three edges: top edge (from (-1,1) to (1,1)), left edge (from (-1,1) to (0, - 1)), right edge (from (1,1) to (0, - 1)). Now, take a vertical line x = a where - 1 < a < 1. The vertical line will intersect the top edge at (a,1) and either the left or right edge? Wait, no, the left edge is for x from - 1 to 0, and the right edge is for x from 0 to 1. So if a is between - 1 and 0, the vertical line x = a intersects the top edge (y = 1) and the left edge (which goes from (-1,1) to (0, - 1)). So that's two intersection points. Similarly, for a between 0 and 1, it intersects the top edge and the right edge. Therefore, the vertical line test fails for Graph 3, so it is not a function. But wait, the original checkboxes: Graph 3 is unchecked. Wait, the question is about whether Graph 3 represents a function. So according to the vertical line test, it does not, so the box for Graph 3 should remain unchecked.

Answer:

The box for Graph 3 should be unchecked (as it does not represent a function, confirmed by the vertical line test).