Polynomial Behaviour

Overview

This lesson focuses on polynomial functions of higher degrees using the key ideas students have developed with cubic functions: identifying features from algebra and sketching graphs (with and without technology). Students will practise how end behaviour and intercept information shape the overall sketch.

Learning intentions

Students will:

• identify key coefficients/degree and describe polynomial form from expressions
• sketch the graph of a polynomial using end behaviour and intercept reasoning
• explain how real zeros relate to x-intercepts (including multiplicities)
• interpret graph features: turning points (qualitatively) and behaviour as (x \to \infty) and (x \to -\infty)

Success criteria

“I can …”

• determine the degree and leading coefficient of a polynomial and state end behaviour
• predict and justify where a polynomial crosses or touches the x-axis using multiplicity ideas
• sketch a reasonable polynomial graph consistent with intercepts, end behaviour, and turning points
• use technology to check a sketch and refine it using graph behaviour evidence

Curriculum links

• Recognise and determine features of the graphs of cubic functions, including shape, intercepts, and behaviour as (x \to \infty) and (x \to -\infty)
• Sketch the graphs of cubic functions with and without technology (adapted to higher-degree polynomials by using the same end-behaviour and intercept reasoning)
• Identify the coefficients and the degree of a polynomial
• Use factor forms by expanding or reasoning from factors (to link algebra and graph features)

Lesson structure (60 minutes)

1. 0–7 min · Warm-up: “End behaviour first”. Teacher displays three leading-term prompts (e.g., (f(x)= -2x^4+ \dots), (g(x)=3x^5+\dots), (h(x)=x^6+\dots)). Students quickly write the expected end behaviour for each and share one justification using “degree parity + sign of leading coefficient”.

2. 7–18 min · Mini teach: from cubics to higher polynomials. Teacher explicitly connects known cubic ideas to higher degree: (i) degree and leading coefficient drive end behaviour; (ii) x-intercepts come from factors; (iii) multiplicity affects crossing vs touching. Students copy a short “decision chain” poster:

• Find degree and leading coefficient → determine (x \to \infty), (x \to -\infty)
• Use factors/zeros → predict x-intercepts
• Use multiplicity → crossing (odd) vs touch/bounce (even)
• Use turning points qualitatively (up to degree − 1) to guide sketch shape

3. 18–33 min · Worked example with discussion. Teacher writes a polynomial in factored form (e.g., (p(x)=a(x+2)^2(x-3)) with (a>0)) and asks students to predict: degree, end behaviour, x-intercepts, and crossing behaviour at each intercept. Students complete a guided table (Intercept, multiplicity, cross/touch, likely sketch direction), then teacher sketches a rough graph on the board and labels key features. Teacher prompts: “Where should the graph flatten at an even multiplicity root?” and “How does the sign of (a) affect the ends?”

4. 33–48 min · Technology check + refinement. Teacher provides (or projects) the same polynomial but with numeric (a) and asks students to graph it using a calculator/graphing tool. Students compare their sketch with the digital graph, then revise: adjust intercept markers, estimate turning points, and confirm end behaviour. Students must note one agreement and one correction in a short response.

5. 48–58 min · Independent practice (single student focused). Teacher gives one new polynomial (teacher-selected for the class context) in either expanded or partially factored form and requests a sketch and explanation:

• state degree and leading coefficient
• determine end behaviour
• identify x-intercepts and cross/touch
• indicate approximate turning points (qualitative) Students submit a completed “sketch + justification” sheet.

6. 58–60 min · Exit ticket. Teacher asks one final prompt: “For a polynomial of degree 5 with a negative leading coefficient, describe the end behaviour as (x \to \infty) and (x \to -\infty).” Students answer in one or two sentences.

Resources

• Whiteboard/markers or digital board
• Decision-chain poster (degree, leading coefficient, end behaviour, intercepts, multiplicity)
• Worked example slides or printed examples
• Graphing calculator or computer-based graphing tool
• Student sketch paper (coordinate grid, labelled axes)
• “Sketch + justification” worksheet
• Exit ticket slip (half page)

Assessment

• Formative during warm-up: teacher listens for correct end-behaviour statements and justifications
• Formative in guided example: table completion accuracy (degree, intercepts, cross/touch)
• Check worksheet at 48–58 minutes: correctness of end behaviour, intercept predictions, and a coherent sketch
• Exit ticket at 58–60 minutes: individual mastery of end-behaviour rule

Differentiation

• Support: provide a partially completed decision-chain and a sentence starter set (e.g., “The leading term is …, so the degree is … which means as (x \to \infty) the graph …”)
• Support: offer an additional example with the same structure but simpler coefficients during the worked example
• Extension (light): ask the student to estimate the number of turning points using the “at most degree − 1” idea and check with technology
• SEN/EAL: allow verbal explanation alongside written justification; use consistent vocabulary (“crosses”, “touches”, “bounces”, “end behaviour”); provide an extra blank table template for intercept/multiplicity