Parabolas: Patterns, Graphs, and Solutions

Overview

Students investigate how quadratic rules create parabolic graphs. They identify the vertex, axis of symmetry, intercepts and solutions, then connect graphical features with factorised algebraic forms using a real-world projectile context.

Learning intentions

Students will:

• recognise quadratic patterns from tables and graphs
• construct and interpret parabolas from quadratic rules
• identify the vertex, axis of symmetry, y-intercept and x-intercepts
• connect x-intercepts with solutions of quadratic equations
• investigate how changing parameters affects a parabola

Success criteria

• I can identify key features of a parabola accurately.
• I can use a table or rule to graph a quadratic function.
• I can explain that the x-intercepts are the solutions of the related quadratic equation.
• I can describe how changing a parameter changes the graph.

Curriculum links

• Quadratic functions and equations: identifying and graphing quadratics, interpreting features, and solving equations graphically and numerically.
• Transformations of related functions: investigating the effects of changing parameters on parabolas using digital tools.
• Algebraic representation: connecting factorised monic quadratics with their x-intercepts.
• Mathematical modelling: interpreting a quadratic graph in a practical context.

Lesson structure (60 minutes)

1. 0–6 min · Engaging launch. Display a basketball shot or water-jet image in the parabola hook and lesson sequence and ask, “Where is the object highest, and when does it reach the ground?” Students make a quick estimate, then share what the curved path might tell us about the situation. Teacher note: Emphasise that many projectile paths are approximately parabolic, while noting that the lesson uses a mathematical model rather than a complete physics model.

2. 6–18 min · Explicit modelling. Use the same slide deck to model (y=x^2-4x+3=(x-1)(x-3)), completing a table for (x=0,1,2,3,4) and plotting the points. Think aloud while identifying the y-intercept ((0,3)), x-intercepts ((1,0)) and ((3,0)), vertex ((2,-1)), and axis of symmetry (x=2). Students annotate their copy of the graph and answer: “Why are the roots 1 and 3?” Teacher note: Explain that the roots are where (y=0), so they are the x-intercepts and solutions of (x^2-4x+3=0). Link the factorised form to the null factor law.

3. 18–28 min · Graph interpretation checkpoint. Reveal three questions on the graph interpretation prompts: a) For (y=-(x-2)^2+5), state the vertex, axis of symmetry and maximum value. b) How many real solutions does (y=x^2+2) have when (y=0)? c) For (y=(x+1)(x-4)), predict the x-intercepts before graphing. Students answer independently, compare with a partner and hold up responses. Answers: a) vertex ((2,5)), axis (x=2), maximum (5); b) none, because the graph has no x-intercepts; c) ((-1,0)) and ((4,0)). Formative check: Address the misconception that the vertex is always the y-intercept and that every quadratic has two real roots.

4. 28–40 min · Guided practice. Distribute the quadratic graphs and practice worksheet. Complete Questions 1–2 together, using the displayed scaffold: “Find the y-intercept by setting (x=0); find x-intercepts by setting (y=0); use symmetry to locate or check the vertex.” Students then complete Question 3 with a partner. Teacher checkpoint: Circulate and ask, “What does (y=0) represent?” and “How can you check the axis of symmetry?” Answers: Q1 for (y=x^2-6x+8): y-intercept ((0,8)), x-intercepts ((2,0)), ((4,0)), axis (x=3), vertex ((3,-1)). Q2 for (y=-x^2+4x+5): y-intercept ((0,5)), x-intercepts ((-1,0)), ((5,0)), axis (x=2), vertex ((2,9)). Q3: students should justify that symmetric x-values have equal y-values.

5. 40–53 min · Differentiated independent work. Students complete one pathway on the differentiated quadratic practice. Support students use a partially completed table and labelled axes to graph (y=x^2-2x-3), then identify its features. Core students graph and interpret (y=-x^2-2x+8), including the realistic domain where (y\geq0). Extension students use optional graphing technology to compare (y=x^2), (y=2x^2), (y=-x^2), and (y=(x-3)^2+2), recording the effect of each parameter change. Answers: Support: x-intercepts ((-1,0)), ((3,0)), y-intercept ((0,-3)), axis (x=1), vertex ((1,-4)). Core: x-intercepts ((-4,0)), ((2,0)), y-intercept ((0,8)), axis (x=-1), vertex ((-1,9)); (y\geq0) for (-4\leq x\leq2). Extension: (2x^2) is narrower, (-x^2) reflects in the x-axis, and ((x-3)^2+2) moves the vertex to ((3,2)).

6. 53–60 min · Exit ticket and review. Students complete the final section of the exit ticket before contributing one response to the plenary prompt, “How does a graph show the solutions of a quadratic equation?” Exit ticket: For (y=(x-2)(x+3)): identify the x-intercepts and explain what they mean; state the axis of symmetry; sketch the shape. Answers: x-intercepts ((2,0)) and ((-3,0)), so the solutions are (x=2) and (x=-3); axis (x=-0.5); upward-opening parabola. Collect tickets to identify students requiring further support with intercepts, symmetry or factorisation.

Resources

• the parabola hook and lesson sequence
• the quadratic graphs and practice worksheet
• Graph paper and rulers
• Pencils, erasers and coloured pens
• Whiteboard and markers
• Optional graphing technology
• Projector or interactive display

Assessment

• Questioning during modelling checks whether students understand that roots occur where (y=0).
• Mini-whiteboard or hand-up responses during graph interpretation reveal errors with vertices, axes and number of solutions.
• The worksheet and exit ticket assess graph construction, interpretation, parameter changes and connections between algebraic and graphical representations.

Differentiation

• Support: provide a completed example, a table template, labelled axes, a feature-matching word bank and paired discussion before independent work.
• EAL learners benefit from visual examples and sentence stems such as “The x-intercepts are ___ because ___” and “The axis of symmetry is ___ because ___.”
• Students requiring adjustments may use enlarged graph paper, a calculator or graphing technology and complete fewer questions with the same essential features.
• Extension: ask students to create a quadratic with x-intercepts of their choice, sketch it, and explain how changing the leading coefficient affects its width and direction.