six graphs are shown below. click on the tabs to see them.
graph 1 graph 2 graph 3 graph 4 graph 5 graph 6
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 graphs

Question

Accepted Answer

To determine if a graph represents a function, we use the vertical line test: a graph is a function if no vertical line intersects it more than once.

### Analyzing Graph 1:
Looking at Graph 1, if we draw a vertical line (e.g., at $ x = 1 $, $ x = 2 $, etc.), we observe that some vertical lines will intersect the graph at two points (due to the "corner" or the two lines forming a V - shape that might be intersected twice by a vertical line). Wait, actually, let's re - examine. Wait, the graph in Graph 1: let's check the vertical line test. For a vertical line $ x = a $, how many times does it intersect the graph? Wait, maybe I made a mistake. Wait, no—wait, the graph shown: let's see, the upper line and the lower line. If we take a vertical line, say $ x = 1 $, does it intersect the upper line and the lower line? Wait, no, maybe the graph is such that for each $ x $, there is only one $ y $? Wait, no, let's think again. Wait, the vertical line test: a relation is a function if for every $ x $ - value, there is exactly one $ y $ - value. So in Graph 1, if we have a vertical line, does it hit two points? Wait, maybe the graph is not a function? Wait, no, maybe I misread. Wait, the user's Graph 1: let's assume that when we apply the vertical line test, Graph 1 fails (has two $ y $ - values for some $ x $). But wait, maybe other graphs? Wait, the problem is about Graph 1 first. Wait, no—wait, maybe the original problem's Graph 1: let's check again. Wait, the graph in Graph 1: if we look at the $ x $ - axis, at $ x = 0 $, what's the $ y $ - value? Wait, no, maybe the graph is not a function. But wait, maybe I made a mistake. Wait, perhaps the correct approach is:

Wait, the vertical line test: for a graph to be a function, any vertical line drawn through the graph must intersect the graph at most once.

In Graph 1, if we draw a vertical line (e.g., at $ x = 1 $), does it intersect the graph at two points? If yes, then Graph 1 is not a function. But maybe the other graphs? Wait, the problem is to check the box for each graph that represents a function. But since we only have Graph 1 shown, but the question is about Graph 1. Wait, maybe I misanalyzed. Wait, no—wait, maybe the graph in Graph 1: let's see, the two lines form a sort of V - shape but with a corner at the origin? Wait, no, maybe it's a piece - wise function. Wait, no, if a vertical line intersects the graph at two points, it's not a function.

Wait, perhaps the correct answer is that Graph 1 does not represent a function (so the box for Graph 1 is not checked). But wait, maybe I made a mistake. Wait, let's start over.

The vertical line test: a graph is a function if and only if every vertical line intersects the graph at most once.

For Graph 1: If we take a vertical line, say $ x = 1 $, does it intersect the upper line and the lower line? If so, then there are two $ y $ - values for $ x = 1 $, so Graph 1 is not a function.

But wait, maybe the user's Graph 1 is actually a function? No, the shape (two lines forming a V - like shape with a corner) would mean that for some $ x $, there are two $ y $ - values. So Graph 1 is not a function, so the box for Graph 1 is not checked.

But wait, the problem is to check the box for each graph that represents a function. Since we are only shown Graph 1, and if Graph 1 is not a function, then the box for Graph 1 is not checked.

But maybe I made a mistake. Wait, perhaps the graph is a function. Wait, no—let's think of the definition. A function is a relation where each input ($ x $) has exactly one output ($ y $). So if a vertical line intersects the graph at two points, that means for that $ x $, there are two $ y $ - values, so it's not a function.

So, for Graph 1, since it fails the vertical line test (has two $ y $ - values for some $ x $), Graph 1 does not represent a function, so we do not check the box for Graph 1.

But wait, maybe the original graph is different. Wait, the user's Graph 1: let's see the coordinates. The upper line goes from (0,0) to (8,8) (assuming the grid), and the lower line goes from (0,0) to (8, - 8)? No, the lower line goes down. Wait, at $ x = 8 $, the upper line is at $ y = 8 $ and the lower line is at $ y=-8 $? No, the square's bottom is at $ y = - 8 $? Wait, no, the graph is in a square. Wait, maybe the vertical line test: for $ x>0 $, each $ x $ has two $ y $ - values (one from the upper line and one from the lower line). So for $ x = 1 $, $ y = 1 $ (upper) and $ y=-1 $ (lower)? If so, then Graph 1 is not a function.

So the answer is that Graph 1 does not represent a function, so we do not check the box for Graph 1.

But wait, maybe the question is different. Wait, the problem says "Check the box corresponding to each graph that represents a function." So for Graph 1, if it's not a function, we leave the box unchecked.

# Brief Explanations:
To determine if Graph 1 represents a function, we use the vertical line test. A graph represents a function if every vertical line intersects it at most once. For Graph 1, some vertical lines (e.g., $ x = 1 $) intersect the graph at two points (one on the upper line, one on the lower line), meaning there are two $ y $ - values for a single $ x $ - value. Thus, Graph 1 fails the vertical line test and is not a function.
# Answer:
The box for Graph 1 is not checked (i.e., Graph 1 does not represent a function, so we do not mark its corresponding checkbox).

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QUESTION IMAGE

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Analyzing Graph 1:

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