Question

six graphs are shown below. click on the tabs to see them.
graph 1 graph 2 graph 3 graph 4
(graph 2s coordinate system and points are shown)
check the box corresponding to each graph that represents a function.
○ graph 1 ○ graph 2 ○ graph 3 ○ graph 4 ○ graph 5

Explanation:

Step1: Apply Vertical Line Test

A graph represents a function if no vertical line intersects it more than once. For Graph 2, each x-value has exactly one corresponding y-value (all plotted points have unique x-coordinates: -4, 1, 3, 5, 5? No, correction: x-values are -4, 1, 3, 5, 5? No, looking at the plot: (-4,5), (1,-4), (3,-3), (5,-2), (5,-1). Wait, no—wait, x=5 has two y-values (-2 and -1). Correction: Wait, no, recheck:
Wait, the points are (-4, 5), (1, -4), (3, -3), (5, -2), (5, -1). So x=5 has two y-values. Wait no, no—wait, the vertical line test: if any vertical line passes through more than one point, it's not a function. But wait, the user only provided Graph 2, but the question asks about all graphs. Wait, no, the user only shows Graph 2, but the problem mentions 6 graphs. Wait, no, the user's image only shows Graph 2. Wait, no, the user's input is the image with Graph 2, and the question is to check boxes for graphs that are functions. But since only Graph 2 is visible:

Wait, correction: For Graph 2, the points are (-4, 5), (1, -4), (3, -3), (5, -2), (5, -1). The x-value 5 has two different y-values, so a vertical line x=5 passes through two points. Therefore, Graph 2 does not represent a function. But wait, no—wait, maybe I misread the points. Let's recheck:
Looking at the grid:

• The top point is (-4, 5) (x=-4, y=5)
• Then (1, -4) (x=1, y=-4)
• (3, -3) (x=3, y=-3)
• (5, -2) (x=5, y=-2)
• (5, -1) (x=5, y=-1)

Yes, x=5 has two y-values, so Graph 2 fails the vertical line test. But since the user only provided Graph 2, and the question asks about all graphs, but only Graph 2 is shown. Wait, no, the user's input is incomplete? No, wait, the user's image shows Graph 2, but the question mentions 6 graphs. But the user only provided Graph 2. Wait, but the task is to check boxes for graphs that are functions. But since only Graph 2 is visible, we can only assess Graph 2.

Step2: Conclude for Graph 2

Since x=5 has two y-values, Graph 2 is not a function.

But wait, maybe I misread the points. Let's recheck:
Wait, the rightmost two points: one is (5, -1), one is (5, -2). So vertical line x=5 intersects two points, so it's not a function.

Answer:

Graph 2 does not represent a function, so its box should NOT be checked.

Wait, but the user's question is to check boxes for graphs that are functions. Since only Graph 2 is visible, we can only evaluate it: Graph 2 fails the vertical line test (one x-value maps to two y-values), so it is not a function.

But wait, maybe I made a mistake. Let's re-express:
The vertical line test states that a graph is a function if every vertical line intersects the graph at most once. For Graph 2, the vertical line $x=5$ intersects the graph at two distinct points $(5, -1)$ and $(5, -2)$, so it violates the vertical line test.

Final Answer:

Graph 2 is not a function, so do not check the box for Graph 2.

But wait, the user's question is to check boxes for graphs that are functions. Since only Graph 2 is provided, we can only conclude that Graph 2 should not be checked.

Wait, but maybe I misread the x-coordinates. Let's check again:
The point at the bottom right: x=5, y=-2? And the one above it: x=5, y=-1? Yes. So x=5 has two y-values. So Graph 2 is not a function.