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 (the provided graph), we observe that there is a vertical line (or a point where a vertical line would intersect the graph more than once, e.g., at $ x = 1 $ or similar, based on the "V" shape with a vertex at $ x = 1 $ or similar). Wait, actually, let’s re - examine: The graph shown has a "corner" at $ x = 1 $ (or the x - value where the two lines meet). Wait, no—wait, the vertical line test: for a graph to be a function, every $ x $ - value must have at most one $ y $ - value. In Graph 1, if we draw a vertical line at $ x = 1 $ (the x - coordinate of the vertex), how many times does it intersect? Wait, no—wait, the graph has two lines: one with a positive slope and one with a negative slope, meeting at a point (the vertex). Wait, actually, no—wait, the vertical line test: if we take any vertical line $ x = a $, how many $ y $’s does it have? For $ x < 1 $ (assuming the vertex is at $ x = 1 $), does the graph exist? Wait, the graph is only for $ x\geq1 $? No, looking at the axes, the graph is drawn such that for $ x = 1 $, there are two $ y $ - values? Wait, no—wait, the graph as shown: the two lines meet at a point (the vertex), so for the $ x $ - value of the vertex, there is one $ y $ - value, and for other $ x $ - values, only one $ y $ - value? Wait, no, I think I made a mistake. Wait, the graph in the image: let's assume the vertex is at $ (1, - 1) $ or something. Wait, no—actually, the key is: a function must have, for each input $ x $, exactly one output $ y $. So if a vertical line intersects the graph more than once, it's not a function.

Wait, maybe the original problem's Graph 1: let's re - evaluate. Wait, the user's graph (Graph 1) has a "V" shape but with the vertex at $ x = 1 $, and for $ x>1 $, two lines? No, no—wait, no, the graph is a piece - wise function? Wait, no, the vertical line test: if we take $ x = 1 $, how many times does the vertical line intersect the graph? If the two lines meet at $ (1, k) $, then it's one point. For $ x>1 $, does the graph have two $ y $ - values? Wait, no, the graph as drawn: one line going up from the vertex and one going down? No, that would mean for $ x>1 $, each $ x $ has two $ y $ - values, which would violate the vertical line test. Wait, maybe I'm misinterpreting the graph.

But perhaps the other graphs (Graphs 2 - 6, which we can't see) are like:

- Graphs that pass the vertical line test (e.g., parabolas opening up/down, lines, etc.) are functions.
- Graphs that fail (e.g., vertical lines, circles, or graphs with multiple $ y $ for one $ x $) are not.

But since the user's question is to check the boxes for graphs that represent functions, and assuming that Graph 1 fails (because it has two $ y $ - values for $ x>1 $, violating the vertical line test), and other graphs:

Wait, maybe the correct approach is:

A graph represents a function if and only if no vertical line intersects the graph more than once (vertical line test).

So, for example:

- A line (non - vertical) is a function.
- A parabola opening up or down is a function.
- A circle is not (fails vertical line test).
- A vertical line is not.
- A graph with two branches (like a hyperbola opening left - right) is not.

Assuming that Graph 1 fails (has two $ y $ for some $ x $), and other graphs:

But since the user's problem is to check the boxes, and maybe the correct answer is that Graph 1 does not represent a function, and some of the other graphs do. But since we can't see Graphs 2 - 6, but maybe in the original problem, the correct graphs that are functions are, say, Graphs 2, 3, 4, 5, 6 (depending on their shapes), but since the user's Graph 1 is the one shown, and it fails the vertical line test (has two $ y $ - values for $ x>1 $), so Graph 1 is not a function.

But maybe I made a mistake. Wait, perhaps the graph is a piece - wise function with the vertex at $ x = 1 $, and for $ x\geq1 $, two lines? No, that can't be. Wait, no—maybe the graph is $ y = |x - 1| - 1 $, which is a V - shaped graph (absolute value function), which is a function (passes vertical line test). Oh! Wait, the absolute value function $ y=|x - h|+k $ is a function, because for each $ x $, there is exactly one $ y $. So maybe I misinterpreted the graph. If the graph is $ y = |x - 1| - 1 $, then it's a function.

So let's re - do the vertical line test for $ y = |x - 1| - 1 $: for any $ x $, plugging in $ x $ gives exactly one $ y $. So a vertical line $ x = a $ will intersect the graph at exactly one point $ (a, |a - 1| - 1) $. So in that case, Graph 1 is a function.

Ah, I see my mistake earlier. The V - shaped graph (absolute value function) is a function because it passes the vertical line test (each $ x $ has one $ y $). So if Graph 1 is an absolute value graph, then it is a function.

So, to summarize:

- A graph is a function if no vertical line intersects it more than once.
- The V - shaped graph (absolute value) passes this test, so Graph 1 is a function.

Assuming other graphs:

- If a graph is a vertical line, it's not.
- If a graph is a circle, it's not.
- If a graph has two $ y $ for one $ x $, it's not.

But since we can't see Graphs 2 - 6, but based on the given Graph 1 (V - shaped, absolute value - like), it is a function.

So the answer would be to check the box for Graph 1 (and other graphs that pass the vertical line test).

But since the user's problem is to check the boxes, and assuming Graph 1 is a function (absolute value graph), we would check Graph 1's box.

# Brief Explanations:
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. For Graph 1 (a V - shaped graph, likely an absolute - value - type function), each $ x $ - value corresponds to exactly one $ y $ - value, so it passes the vertical line test and represents a function. (For other graphs, we would apply the same test, but since only Graph 1 is shown in detail and it passes the test, we check its box.)
# Answer:
Check the box for Graph 1 (i.e., mark the checkbox next to "Graph 1"). For other graphs, apply the vertical line test: check the box for a graph if no vertical line intersects it more than once.

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QUESTION IMAGE

Question

Explanation:

Analyzing Graph 1:

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