Question 2
The total number of vertices in this graph is
Not quite. 3 is the number of faces, the regions the graph cuts the page into: the sliver between the straight and curved edges, the big region below the curve, and the outside. One way to get 3 is to count regions instead of dots. In 2022, 0.6% of Queensland students chose (A).
Not quite. One way to get 5 is to skip the dot at the top left. It only has one edge, so it looks like the end of a line, but every edge has a vertex at each end. It counts. In 2022, 2.1% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2022, 91.8% of Queensland students chose (C).
Not quite. 7 is the number of edges, the lines. The question asks for vertices, the dots. The curved edge is an extra edge between two dots that are already there, so it adds to the edges but not the vertices. In 2022, 5.0% of Queensland students chose (D).
Every option is a real count from this graph. Pick what to count and the graph numbers them for you.
In a drawn graph, the vertices are the dots and the edges are the lines joining them. Straight or curved makes no difference.
Every edge has a vertex at each end. So the dot at the top left, with only one edge, is still a vertex. Its degree is 1.
Count the dots in columns so you don’t miss one or count one twice.
- Left side: top, middle, bottom. That’s 3.
- Middle: top and bottom. That’s 2.
- Far right: 1.
$$3+2+1=6\text{ vertices}$$
The curved edge starts and ends at dots you have already counted, so it adds nothing here.
Check with Euler’s formula for a connected planar graph. This one has 7 edges and 3 faces, counting the outside:
$$v+f-e=6+3-7=2\ \checkmark$$
That also explains the wrong options: 7 is the edges, 3 is the faces, and 5 is what you get if the dead end at the top left goes missing.
Count the dots: three down the left side, two in the middle (top and bottom) and one on the far right.
$$3+2+1=6\text{ vertices}$$
The single-edge dot at the top left counts, and the curved edge adds an edge, not a vertex. Euler’s formula checks it: $6+3-7=2$.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2022 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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