Review of Functions and Their Properties

Active learning works for this topic because students must repeatedly connect algebraic expressions to visual graphs and apply precise definitions of domain, range, and transformations. Moving between representations strengthens conceptual fluency more than passive review. The relay, stations, and matching tasks provide the spaced practice needed to resolve common confusions about asymptotes and intercepts.

• Year 12
• Mathematics
• 4 activities
• 30 min–50 min

Learning Objectives

1. Compare the graphical and algebraic characteristics of polynomial, rational, exponential, and logarithmic functions.
2. Analyze the impact of transformations (shifts, stretches, reflections) on the equations and graphs of various function types.
3. Construct a function, specifying its type and parameters, that meets a given set of graphical and algebraic properties.
4. Explain the end behavior and monotonicity of different function families based on their algebraic form.
5. Identify the domain, range, intercepts, and asymptotes for a given function from its equation or graph.

Concept Mapping

⏱ 35 min·Pairs

Graph Matching Relay: Function Properties

Prepare cards with graphs, equations, and property lists for polynomial, rational, exponential, and logarithmic functions. Pairs match sets at their desk, then one student relays to a class board to post. Switch roles and discuss mismatches as a group.

Prepare & details

Compare and contrast the key characteristics of different function families.

Facilitation Tip: For the Graph Matching Relay, assign mixed-function pairs so students discuss growth and decay across families, not just within one type.

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Concept Mapping

⏱ 45 min·Small Groups

Transformation Chain: Sequential Changes

Small groups receive a base function graph and a sequence of transformation cards, like 'stretch vertically by 2, shift right 3'. They sketch each step on grid paper and predict the final equation. Groups share and verify with graphing software.

Prepare & details

Analyze how transformations affect the graphs and equations of various functions.

Facilitation Tip: During Transformation Chain, have each student write the next equation and graph on a mini-whiteboard before passing it on to prevent silent copying.

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Concept Mapping

⏱ 50 min·Small Groups

Function Builder Stations: Property Challenges

Set up stations with property cards, such as 'vertical asymptote x=1, horizontal y=0, passes through (0,2)'. Small groups construct matching rational or exponential equations, graph them, and test with tables of values. Rotate stations twice.

Prepare & details

Construct a function that satisfies a given set of graphical and algebraic properties.

Facilitation Tip: At Function Builder Stations, require students to state their function’s family and key properties aloud before moving to the next challenge.

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Concept Mapping

⏱ 30 min·Pairs

Comparison Matrix: Family Contrasts

Whole class starts with a shared digital or printed matrix comparing function families. In pairs, students fill rows for characteristics like end behavior, then contribute one unique insight per pair to the class matrix via sticky notes.

Prepare & details

Compare and contrast the key characteristics of different function families.

Facilitation Tip: In the Comparison Matrix, provide colored pencils so students can code intercepts, asymptotes, and intervals of increase or decrease consistently.

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Teaching This Topic

Teachers approach this topic by anchoring every concept in graph-algebra translation. Avoid teaching families in isolation; instead, sequence examples that force comparison of similar transformations across families. Use the whiteboard to sketch quick graphs as students describe transformations, making abstract shifts concrete. Research shows that students solidify understanding when they generate examples that meet constraints, so prioritize construction tasks over recognition tasks.

What to Expect

Successful learning looks like students accurately matching graphs to equations, describing transformations in sequence, and constructing functions that meet specified conditions with clear reasoning. They should justify choices using domain, range, and asymptote language, showing they can analyze function families side by side. End behavior and monotonicity should be part of every explanation.

Watch Out for These Misconceptions

Common Misconception

During Graph Matching Relay, watch for students assuming all exponential functions grow without bound.

What to Teach Instead

Have students plot three points on each exponential graph in pairs, comparing functions with bases 2, 1/2, and 10 to reveal decay as well as growth patterns.

Common Misconception

During Graph Matching Relay, watch for students assuming rational functions never cross vertical asymptotes.

What to Teach Instead

Ask students to evaluate the function at x-values just left and right of the vertical asymptote using tables of values to observe sign changes and behavior limits.

Common Misconception

During Function Builder Stations, watch for students assuming logarithmic functions accept negative inputs.

What to Teach Instead

Require students to write the domain explicitly next to each constructed function and verify it by testing an invalid input, using the station’s graphing tool to confirm the undefined region.

Assessment Ideas

Quick Check

After Graph Matching Relay, collect student answer sheets and circulate to check correct matches and identified function families. Use a simple rubric: 1 point per correct match and 1 point per accurate family label.

Exit Ticket

During Transformation Chain, ask students to write their final function on a slip and explain how the sequence of transformations produced its end behavior. Collect these to assess understanding of cumulative changes.

Discussion Prompt

After Comparison Matrix, facilitate a class discussion where groups present contrasts between a polynomial and an exponential function with similar transformations. Listen for precise language about growth rates and asymptotes to gauge depth of understanding.

Extensions & Scaffolding

• Challenge: Ask students to design a function that has a hole at x=4 and a vertical asymptote at x=1, explaining why the hole remains after factoring.
• Scaffolding: Provide pre-labeled axes with key points and asymptotes for students to sketch transformations step-by-step using a T-chart.
• Deeper: Have students research and present real-world phenomena modeled by each function family, linking parameters to contextual meaning.

Key Vocabulary

Asymptote	A line that a curve approaches arbitrarily closely. Vertical asymptotes occur where a rational function's denominator is zero, and horizontal asymptotes describe end behavior.
Monotonicity	Describes whether a function is consistently increasing or decreasing over an interval. Polynomials can change monotonicity, while exponential and logarithmic functions are strictly monotonic.
End Behavior	The behavior of a function's graph as the input (x) approaches positive or negative infinity. This is determined by the leading terms of polynomials or the base of exponential/logarithmic functions.
Transformation	Operations applied to a function's equation or graph, such as translations (shifts), dilations (stretches/compressions), and reflections, which alter its position or shape.