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 1 image: a coordinate plane with several points plotted
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 the points in Graph 1, we check for repeated $ x $-values (since a vertical line at a specific $ x $-value should only hit one point). The $ x $-coordinates of the points are $ -4 $, $ -3 $ (wait, no—let’s list the coordinates:
- $ (-4, 1) $ (assuming the grid: $ x=-4 $, $ y=1 $)
- $ (0, 3) $ ( $ x=0 $, $ y=3 $)
- $ (1, 5) $ ( $ x=1 $, $ y=5 $)
- $ (5, 0) $ ( $ x=5 $, $ y=0 $)
- $ (-3, -5) $ ( $ x=-3 $, $ y=-5 $)
- $ (-1, -5) $ ( $ x=-1 $, $ y=-5 $)

Wait, $ x=-3 $ and $ x=-1 $ each have one point, $ x=-4 $, $ x=0 $, $ x=1 $, $ x=5 $ each have one point. Wait, no—wait, the two points at the bottom: $ x=-3 $ (maybe $ x=-3 $) and $ x=-1 $? Wait, no, looking at the grid: the bottom two points—let’s check their $ x $-coordinates. Suppose the left bottom point is at $ x=-3 $, $ y=-5 $, and the right bottom point is at $ x=-1 $, $ y=-5 $. So each $ x $-value has only one $ y $-value. Wait, but wait—wait, the vertical line test: for any $ x $, is there only one $ y $? Let’s check $ x=0 $: only one point ( $ 0, 3 $ and $ 0, 5 $? Wait, no—wait the graph: there are two points on the $ y $-axis? Wait, the original graph: “Graph 1” has a point at $ (0, 3) $ (since $ y=3 $ is between 2 and 4, maybe $ y=3 $) and a point at $ (0, 5) $? Wait, no, the user’s graph: “Graph 1” has two points on the $ y $-axis ( $ x=0 $): one at $ y=3 $ (or $ y=3 $) and one at $ y=5 $ (above $ y=4 $)? Wait, no—let me re-examine.

Wait, the graph shows:
- A point at $ x=-4 $, $ y=1 $ (approx)
- A point at $ x=0 $, $ y=3 $ (middle vertical line, $ y=3 $)
- A point at $ x=0 $, $ y=5 $ (middle vertical line, $ y=5 $)
- A point at $ x=5 $, $ y=0 $
- A point at $ x=-3 $, $ y=-5 $
- A point at $ x=-1 $, $ y=-5 $

Ah! Here’s the key: at $ x=0 $, there are two points ( $ (0, 3) $ and $ (0, 5) $ ). So a vertical line at $ x=0 $ would intersect the graph twice. Therefore, Graph 1 **fails** the vertical line test and is not a function.

But wait—since we only have Graph 1 visible, but the problem has six graphs. However, the user’s question is to check which graphs represent functions. Since only Graph 1 is shown, but we need to infer. Wait, no—maybe the original problem (but the user provided Graph 1). Wait, perhaps the user made a mistake, but assuming Graph 1: since it has two points with the same $ x $-value ( $ x=0 $ ), it is not a function. But if other graphs (not shown) pass the vertical line test, but since we only have Graph 1, let’s proceed.

But wait—maybe I misread. Let’s re-express:

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

For Graph 1:
- Check $ x=0 $: two points (different $ y $-values). Thus, vertical line $ x=0 $ intersects twice. So Graph 1 is **not** a function.

But since the problem has six graphs, but only Graph 1 is shown, we can’t analyze the others. However, the user’s question is to check the box for each graph that is a function. Since Graph 1 fails the vertical line test (two points on $ x=0 $), Graph 1 is not a function.

But wait—maybe the user intended to ask about Graph 1. Let’s confirm:

If a graph has two points with the same $ x $-coordinate (different $ y $-coordinates), it is not a function. In Graph 1, there are two points at $ x=0 $ (different $ y $-values), so it fails the vertical line test. Thus, Graph 1 is not a function.

But since the problem has multiple graphs, and we can only see Graph 1, we can’t answer fully. However, assuming the question is about Graph 1:

# Brief Explanations:
To determine if a graph represents a function, use the **vertical line test**: a graph is a function if no vertical line intersects it more than once. For Graph 1, there are two points with $ x = 0 $ (different $ y $-values), so a vertical line at $ x = 0 $ intersects the graph twice. Thus, Graph 1 fails the vertical line test and is not a function.

# Answer:
Graph 1: $\boldsymbol{\square}$ (unchecked, since it is not a function)
(Note: For other graphs, we need to analyze them similarly, but only Graph 1 is visible here.)

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QUESTION IMAGE

Question

Explanation:

Analyzing Graph 1:

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