Transformations and Modelling

Overview

In this seventh lesson of the unit Exploring Non-Linear Relationships, students connect algebraic rules with graphical transformations, then use visual evidence from data to select and justify quadratic, exponential or reciprocal models. Students build on prior work with non-linear graphs, equations and digital graphing tools.

Learning intentions

Students will:

• recognise translations, reflections, stretches and dilations of non-linear graphs
• describe how changes to an equation affect its graph
• select a suitable quadratic, exponential or reciprocal model for a data set
• justify a model using the shape, trend and context of the data
• use digital tools to test and refine conjectures.

Success criteria

• I can describe the effect of changing an equation such as (y=f(x)+k), (y=f(x-h)), (-f(x)) or (af(x)).
• I can match a transformed equation to its graph and explain my reasoning.
• I can identify whether data is better represented by a quadratic, exponential or reciprocal model.
• I can justify a model and identify one limitation or assumption.

Curriculum links

• Algebra — experiment with functions and relations using digital tools, make and test conjectures, and generalise emerging patterns.
• Algebra — recognise the connection between algebraic and graphical representations of exponential relations.
• Algebra — choose and evaluate linear, quadratic or exponential models in applied situations, interpreting solutions and reporting assumptions and findings.
• Statistics — construct and interpret graphical representations of relationships between numerical variables, commenting on patterns and association.

Lesson structure (45 minutes)

1. 0–5 min · Hook and retrieval. Open with the hook and retrieval slides showing the base graph (y=x^2) beside an unfamiliar transformed graph, and ask, “What changed, and how can you prove it?” Students independently annotate two visible changes, then compare answers with a partner. Briefly revisit domain, range, vertex and asymptote language where relevant.

2. 5–14 min · Explicit teaching. Use the transformation teaching slides to model the transformation family (y=f(x)+k), (y=f(x-h)), (y=-f(x)), (y=af(x)) and (y=f(bx)), using (f(x)=x^2), (f(x)=2^x) and (f(x)=1/x). Teacher emphasises that horizontal changes appear inside the bracket and demonstrates one point moving from the original graph to the new graph. Students complete a quick “equation change → graph change” response after each example.

3. 14–23 min · Guided digital investigation. Students work in pairs with a graphing tool and the transformations and modelling worksheet. They enter a base function, change one parameter at a time, and record the resulting movement, reflection or stretch in a table. Teacher circulates, checking the common error that (f(x-h)) moves right by (h), not left. Students make one conjecture, test it with a second function, and record whether it holds.

4. 23–34 min · Model selection task. Return to the model-selection example slides and demonstrate how to inspect a data table or scatterplot: a turning pattern may suggest a quadratic, approximately constant multiplicative change may suggest an exponential model, and values changing inversely with (x) may suggest a reciprocal model. In pairs, students use the data sets on the worksheet to choose a model, sketch or generate it digitally, and complete the sentence: “We selected ___ because ___; however, this model may be limited because ___.”

5. 34–41 min · Justify and critique. Display the discussion prompts on the justification and critique slides. Pairs exchange one completed model claim with another pair, which checks whether the evidence supports the choice and asks one challenge question: “Could another model fit?” or “Does the model imply causation?” Students revise their explanation using specific features such as turning point, rate of increase, asymptote, residual pattern or contextual reasonableness.

6. 41–45 min · Exit assessment and preview. Use the plenary and exit-ticket slides to display three final questions. Students submit responses on the worksheet: describe the transformation from (y=f(x)) to (y=-2f(x-3)+1); identify the most suitable model for a data pattern that increases by roughly the same percentage; and state one reason a fitted model should not be trusted outside the observed data range. Preview the final unit lesson, where students will present and evaluate a complete model.

Resources

• the complete transformation and modelling slide deck
• the transformations and modelling worksheet
• Student laptops or tablets with graphing software
• Projector or interactive display
• Exercise books and graphing pencils
• Prepared quadratic, exponential and reciprocal data sets within the worksheet
• Calculators, if required

Assessment

• During explicit teaching, check responses to identify confusion between vertical and horizontal transformations, particularly the direction of (f(x-h)).
• Review pair investigations for accurate graphing, a tested conjecture and evidence-based model selection.
• Use the exit assessment to determine whether students can interpret transformations and distinguish suitable model types. Retain responses for grouping and targeted revision in lesson 8.

Differentiation

• Support students with a transformation reference strip on the worksheet, colour-coded input/output tables, partially completed graphs and sentence starters such as “The graph moves ___ because ___.”
• Pair students strategically and provide teacher-selected data sets with clear patterns before offering less obvious examples.
• For EAL/D and students requiring additional support, pre-teach “translate”, “reflect”, “stretch”, “dilate”, “asymptote”, “rate” and “model” with visual examples; allow verbal justification before written recording.
• Extend confident students by asking them to compare two plausible models, test both digitally, and explain which is more reliable for interpolation and which claims would be unsafe for extrapolation.