Function Transformation Effects

Overview

Students explore how translations, reflections, stretches and compressions change graphs of functions, building on earlier work with transformation ideas and coordinate interpretation. The lesson emphasises identifying a transformation from an equation and predicting its effect on key points and the graph.

Learning intentions

Students will be able to:

• WALT describe how translations, reflections, vertical stretches/compressions, and horizontal stretches/compressions affect function graphs.
• WALT use function form to predict transformations of (f(x)) into (g(x)) (including using parameter “sign and magnitude” reasoning).
• WALT apply transformations to transform selected points and sketch the resulting graph accurately.
• WALT check predictions by comparing transformed key features (intercepts, turning points, symmetry).

Success criteria

Students can:

• I can state what each type of transformation does to a graph (direction and scale).
• I can identify the transformation(s) in a given equation and apply them in the correct order.
• I can transform key points correctly (e.g. intercepts/vertex for quadratics) and sketch a matching graph.
• I can verify my sketch by using features of the transformed equation.

Curriculum links

• Matrices and transformations (QCAA Specialist Mathematics, Unit 2 / Topic 5) — representations of transformations and applying transformations to points and polygons.
• Transformations in the plane — using linear transformation thinking to determine combined effects and interpret results on coordinates.
• Apply transformations to polygons/points — using the same idea of “transform key points then sketch” for function graphs.

Lesson structure (60 minutes)

1. 0–6 min · Warm-up (spot the change). Teacher shows two simple graphs (e.g. a parabola and its transformed version) and asks students to name what changed (shift, flip, stretch/compress). Students answer individually, then share one idea with teacher prompting.

2. 6–15 min · Direct teach (translation, reflection, stretch/compress). Teacher models four transformations on a base function (use a chosen base like (y=x^2) or (y=f(x)) with clear features), showing:

• translations ((x,y)\rightarrow(x+h,y+k)) and ((x,y)\rightarrow(x-h,y-k)) via equation form
• reflections about the (x)-axis, (y)-axis (and when applicable about the origin)
• vertical and horizontal stretches/compressions by factors Students complete a short “match the equation to the transformation” set (teacher checks the first two together, then students continue).

3. 15–26 min · Worked example (from equation to sketch). Teacher uses one example equation with multiple transformations (kept accessible, e.g. a quadratic transformed by a shift and a vertical scale), and explicitly states a method:

• identify the base function
• list transformations in order using the equation structure
• transform key points: (x)-intercepts and the vertex/turning point (or symmetry line), then sketch Students mirror the steps on their own mini-whiteboard or sheet for a second example while teacher circulates and checks at least two calculated key points.

4. 26–42 min · Guided practice (transform key points, then sketch). Teacher provides 3 graphing prompts. For each prompt, students:

• compute images of 3–4 key (x)-values (chosen by teacher: e.g. vertex plus symmetric points)
• sketch the transformed curve using these points and correct shape
• label key features (vertex/axis/intercepts) on the sketch Teacher gives real-time feedback focusing on common errors: wrong sign for shifts, mixing vertical vs horizontal factors, and applying reflection incorrectly.

5. 42–53 min · Combined transformations check (reasoning + quick verification). Teacher introduces one “verification” task: students predict the transformation result for a chosen point (e.g. the vertex point), then verify by substituting (x) into the transformed function equation to confirm (y). Students complete this for one prompt and state whether their sketch matches the calculated feature.

6. 53–60 min · Exit ticket (individual, concise). Students answer two questions:

• Q1: Identify transformations in an equation for (g(x)) written in terms of (f(x)) (or an explicit basic function).
• Q2: Transform one key point correctly and give the new coordinate. Teacher collects immediately to inform next lesson.

Resources

• Printed worksheet with 2 examples and 3 guided practice prompts
• Grid paper or graph paper (A4)
• Mini-whiteboards or lined paper for point transformation working
• Base-function reference sheet (one page showing (f(x)=x^2) key features, and a reminder list of transformation rules)
• Timer (for transitions) and whiteboard/visualiser if available

Assessment

• Formative checks during warm-up naming of transformations (teacher listens for correct direction/sign language).
• During guided practice, teacher checks correctness of transformed key points (not just final sketch).
• Exit ticket: accuracy in identifying transformations and transforming one key point coordinate.

Differentiation

• Support: provide sentence starters such as “The term ((x-h)) causes a shift of …” and a transformation rules table on the worksheet.
• Support: reduce sketch load by limiting to transforming exactly 3 key points for each prompt initially.
• Extension: if students complete early, ask them to explain the transformation using coordinate language (describe how a point ((x,y)) maps to ((x',y'))).
• EAL/SEN considerations: emphasise consistent vocabulary (translation, reflection, stretch, compression) and allow working with a partially completed table of transformations (factor and sign columns).