Parabolas Today

Overview

Students explore what makes a quadratic function a parabola by moving between tables of values, graphs, and quadratic equations. They then connect solutions to the graph’s intercepts, using digital graphing to check results.

Learning intentions

Students will:

• identify quadratic patterns in tables (using second differences as a clue)
• graph quadratic functions and interpret key features such as symmetry and turning points
• determine quadratic equation solutions by relating them to the graph’s x-intercepts
• solve selected quadratic equations numerically and (where possible) algebraically for integer roots

Success criteria

Students can:

• explain why a table represents a quadratic when the second difference is constant
• sketch and graph a parabola accurately using technology and appropriate scale
• state the turning point and axis of symmetry from the graph
• identify solutions as the x-values where the graph crosses the x-axis

Curriculum links

• AC9M9A04: identify and graph quadratic functions; solve quadratic equations graphically and numerically; relate roots to x-intercepts; solve monic quadratic equations with integer roots algebraically
• AC9M9A04: interpret graph features (symmetry, turning point, maximum/minimum) and decide when values lie within a given range
• AC9M9A04: use graphs to determine solutions and recognise “no x-intercepts means no real solutions”

Lesson structure (50 minutes)

1. 0–5 min · Hook (parabola challenge). Teacher shows two quick graphs: one quadratic and one not, asks “Which one is a parabola and why?” Students write a one-sentence justification before discussing in pairs.

2. 5–12 min · Direct teach: quadratic from table patterns. Teacher projects a table from a quadratic rule and walks through first and second differences, highlighting that constant second differences indicate a quadratic. Students complete a second-difference check on a new table in their workbook (no calculator).

3. 12–20 min · Explore with digital graphing (build the parabola). Teacher models entering a quadratic rule into graphing software and adjusting window/scale to see the full shape, then demonstrates reading symmetry and the turning point. Students (in pairs) graph two provided quadratic rules and record:

• turning point (approximate)
• axis of symmetry
• whether the parabola opens up or down

4. 20–30 min · Intercepts = equation solutions. Teacher explains: “For a quadratic function, the roots of the equation are exactly the x-values where the graph crosses the x-axis.” Students choose one parabola and use the graph’s intercept tool (or trace) to estimate x-intercepts, then write the corresponding equation solutions as ordered pairs (x, 0).

5. 30–40 min · Numerical solving check. Teacher sets a short task: “Use your graph to find solutions to this quadratic equation (round to 2 decimals).” Students solve one quadratic equation graphically (from the same parabola) and then confirm their approximations by using the software’s value/equation/solve feature (or by sampling around the intercepts).

6. 40–47 min · Algebra connection (only if integer roots). Teacher selects one monic quadratic with integer roots and factors it (or guides students to factor using the intercepts they found). Students write the factorised form and the algebraic solutions, then compare to their graph/intercept values.

7. 47–50 min · Exit ticket (quick check). Students answer two prompts on a half page:

• Identify if a given table is quadratic and justify using second differences.
• From a provided graph screenshot, state the x-intercepts (solutions) and the turning point type (max/min).

Resources

• Graphing software (computer/tablets) with function graphing and intercept/trace tools
• Printed table-and-graph worksheet (2 tables + 3 quadratic rules + 2 equation prompts)
• Graph paper or digital notebook for sketches and recording turning points
• Data capture template: “Turning point / Axis of symmetry / Intercepts / Solutions”
• Teacher slides with example tables, second differences steps, and graph screenshots
• Coloured pencils (optional) to mark intercepts on printed graphs

Assessment

• Formative checks during table work: observe students’ second-difference calculations and listen for correct reasoning (“constant second difference implies quadratic”).
• Formative checks during graphing: circulate to confirm correct scaling, correct turning point identification, and correct reading of intercepts.
• Exit ticket: check accuracy of solutions from x-intercepts and correctness of the quadratic-table justification.

Differentiation

• Support:
• Provide a partially completed second-difference table for one of the tasks.
• Give sentence starters: “The second difference is constant because…”, “The solution is the x-value where…”.
• Offer a “graph reading checklist” (scale, symmetry, turning point, intercepts).
• Extension:
• Ask students to estimate the y-value at the turning point and state whether it matches the rule when substituted (without heavy algebra).
• For students ready: challenge them to predict intercepts from factors before checking with the graph.
• EAL/SEN considerations:
• Use visual cues (arrows and colour coding) to label “x-intercepts” and “solutions”.
• Keep instructions short and repeat key phrases; allow working with a partner to verbalise reasoning before writing.
• Manage misconceptions:
• Address that “constant first difference” suggests linear, not quadratic.
• Re-emphasise that solutions correspond to x-intercepts only (where y = 0).