"Can you explain the distinctions between graphs, diagrams, and charts, and provide definitions for each of these concepts? Specifically, is every graph considered a diagram? Are graphs exclusively reserved for functions? Is it accurate to refer to an ellipse drawing as a graph, diagram, or chart?"
At the elementary school, level graphs include: bar graphs, pie graphs, picture graphs, line graphs, and histograms. Only the line graphs represent functions. For example, a bar graph might show favorite colors, where 5 children like pink, 5 children like turquoise, and 2 children like green. This a relation that is not a function. Similarly, high school students who look at graphs on the x-y axis are told to use the vertical line test to determine whether a graph shows a function. Many show a relation that is not a function.
Diagrams at the elementary school level include simple geometric figures including a circle, square, etc. They might also include a simple street map where the student has to figure out the shortest path from A to B. At this level we would not call a chart a diagram. This research paper asked secondary school teachers what diagrams are and what are they used for. They concluded:
Diagrams are succinct, visual representations of real-world or mathematical objects that serve to communicate properties of the objects and facilitate problem solving. This study explored perceptions of mathematics teachers in Singapore, who are heads of mathematics departments in their respective schools, related to diagrams and their use in the teaching of mathematics. An open-ended survey was adopted to illicit responses to i) what is a diagram, and ii) when do you use diagrams in your mathematics instruction. The findings of the study show that participants generally viewed diagrams as visual representations of real life or mathematical objects and they used them mainly as visual aids when illustrating mathematical concepts and relationships.
Charts at the elementary level would include a tally chart, such as a chart used to make a bar graph (number of votes for a favorite color in tallies next to each name). Charts include various representations of data including graphs. Pie graphs are called pie charts.
Tables with function values are not graphs but represent functions. They would be called charts as well.