4.4 Graph Interpretation and Model Choice

Model choice is evidence, not a guess

Math 30-2 graph interpretation questions often combine several skills: reading intercepts, recognizing restrictions, interpreting rates of change, choosing a model, and explaining the choice. The current Alberta bulletin says Relations and Functions receives the largest content emphasis, so model-choice questions deserve careful practice. A correct answer is not just the model name; it is the model name plus evidence.

Start by asking what information is available. A graph gives shape, intercepts, maxima, minima, asymptotes, and domain. A table gives patterns in values. An equation gives structure. A context tells you what inputs make sense and whether the output can be negative, fractional, or continuous.

Fast evidence table

This table is a decision aid, not a substitute for context. A table with only three points can fit many models, so use all given information.

Reading graph features

When a graph is provided, label the axes first. Then identify domain, range, intercepts, intervals of increase or decrease, and any maximum or minimum. If there is an asymptote, state it as a line such as x = 4 or y = -2. If the graph has an open circle, that point is excluded even if the curve appears to pass through it.

For polynomial graphs, look at end behaviour and turning points. A quadratic has one turning point. A cubic can have two. Higher-degree polynomials can have more, but the graph must still match the leading coefficient and degree clues. For rational graphs, look for breaks, holes, and branches that approach lines.

Worked example: choosing from a table

A table shows the number of bacteria after equal time intervals:

The first differences are 12, 24, 48, and 96, so they are not constant. The second differences are not constant either. The ratios are 24/12 = 2, 48/24 = 2, 96/48 = 2, and 192/96 = 2. Equal input steps multiply the output by the same factor, so an exponential model is appropriate.

A complete response might say: An exponential model fits because the amount doubles each equal time interval. If t represents time intervals after the first measurement, a model such as A = 12(2)^t matches the table over the observed domain.

Worked example: choosing from graph features

A graph has two branches. The curve approaches x = 5 but never crosses that vertical line, and it levels off near y = 2 for large positive and negative x-values. Those features strongly suggest a rational function with a vertical asymptote x = 5 and a horizontal asymptote y = 2. A polynomial model would not have a vertical asymptote, so it is not appropriate.

Written-response quality

The written-response component of Math 30-2 expects students to solve, explain, justify, and use appropriate units and rounding when needed. For model choice, the strongest responses follow this pattern:

1. Name the model family.
2. Give numerical or graphical evidence.
3. Connect the evidence to the context.
4. State any domain or range restrictions.
5. Write a clear conclusion in a sentence.

If technology is used, say what was entered or compared. A statement like "my calculator said quadratic" is weak. A statement like "the second differences are constant, and the scatterplot has one turning point, so a quadratic model fits the data over the measured interval" is much stronger.

Diploma traps

• Choosing a model from one visible feature while ignoring another, such as selecting polynomial even though the graph has a vertical asymptote.
• Extrapolating far outside the data without saying the prediction may be unreliable.
• Forgetting that a graphing window can hide intercepts, holes, or asymptotes.
• Treating correlation or visual fit as proof that one variable causes another.
• Reporting an answer without units when the context asks for money, time, height, area, or population.

Final check before answering

Before selecting a final model, compare at least two pieces of evidence. A polynomial choice might use end behaviour and turning points. A rational choice might use denominator restrictions and asymptotes. A table choice might use differences or ratios plus the real context. This habit protects you from attractive but incomplete answer choices.

When multiple models seem possible

Short data sets can be misleading. Three points can fit a line, a quadratic, an exponential curve, or many other equations depending on the method used. On the diploma, the best answer usually comes from the pattern the question highlights: equal differences, equal ratios, an asymptote, a turning point, or a stated real-world constraint. If the prompt includes a graph and a table, use both.

When writing a justification, avoid vague phrases such as "it fits best." Replace them with observable evidence: constant second differences, doubling values, a restricted input, a visible asymptote, or a repeating cycle.