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Quadratic Functions: The Parabola

Students explore quadratic functions (y = ax² + bx + c) and their U-shaped graphs — parabolas — discovering how each coefficient shapes the curve, where the vertex is, and why parabolas model everything from thrown balls to satellite dishes.

Grade 9Quadratics50 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Recognize a quadratic function.
  • ✓Graph a parabola.
  • ✓Explain how a affects the shape.
  • ✓Find the vertex.
Essential Question

A thrown ball, a fountain’s arc, a satellite dish — all follow the same U-shaped curve. What is a parabola, and what controls its shape?

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Model · Interactive Tool

The Parabola Grapher

Project this and change a, b, and c. Watch the parabola open up or down, widen or narrow, and see how the vertex moves.

📈 Change a, b, c — shape the parabolaTry it
a=1
b=0
c=-2

The Lesson · Gradual Release (I Do · We Do · You Do)

50 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — The U-Shape

5 min

A ball’s path and a dish’s curve — same shape. What is it?

👩‍🏫 Teacher Moves

  • Show real parabolas.
  • Ask what shape they share.
  • Set the goal: quadratics.

🎒 Student Actions

  • Spot the U-shape.
  • Name it.
  • Predict the equation.
2

I Do — y = ax² + bx + c

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Introduce the quadratic form.
  • Graph a basic parabola.
  • Show a controls opening and width.

🎒 Student Actions

  • Learn the form.
  • Graph one.
  • Connect a to shape.
3

We Do — Graph Together

13 min

Class uses the Parabola Grapher.

👩‍🏫 Teacher Moves

  • Change a, b, c; observe.
  • Find the vertex (turning point).
  • Note up vs. down.

🎒 Student Actions

  • Change the coefficients.
  • Find the vertex.
  • Note the direction.
4

You Do — On Your Own

12 min

Students graph parabolas.

👩‍🏫 Teacher Moves

  • Assign quadratics to graph.
  • Find vertices and direction.
  • Compare two parabolas.

🎒 Student Actions

  • Graph quadratics.
  • Find the vertex.
  • Compare them.
5

Close — Close

5 min

One parabola.

👩‍🏫 Teacher Moves

  • Give a quadratic; describe its graph.
  • State the vertex and direction.
  • Hand out the exit ticket.

🎒 Student Actions

  • Describe the graph.
  • State the vertex.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School: Algebra I).

CCSS
F-IF.7a

Graph quadratic functions and show intercepts, maxima, and minima.

CCSS
A-SSE.3

Produce equivalent forms of a quadratic to reveal properties.

CCSS
F-IF.4

Interpret key features of graphs (intercepts, maximums, minimums).

CCSS
A-APR.3

Identify zeros of polynomials and use them to sketch a graph.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Label the vertex on a graph.
  • Sentence frame: “When a is ___, the parabola opens ___.”
  • Trace real parabolas.

Support & Access

IEP / 504
  • Start with y = x².
  • Change one coefficient at a time.
  • Provide a labeled parabola.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Find the roots (x-intercepts).
  • Rewrite in vertex form.
  • Model a real projectile.
Materials

What to gather

  • 📽️Projector / board
  • 📓Math notebooks
  • 💻The Parabola Grapher
  • 📐Graph paper
  • 🧮Graphing calculators
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

quadratic functiona function with an x² termparabolathe U-shaped graph of a quadraticvertexthe turning point of a parabolaaxis of symmetrythe line splitting a parabola in halfcoefficienta number multiplying a termmaximumthe highest pointminimumthe lowest pointrootan x-intercept (where y = 0)
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
What shape is the graph of a quadratic function?
A parabola (a U-shape).
QUESTION 2
In y = ax² + bx + c, what happens to the parabola when a is negative?
It opens downward (like an upside-down U).
QUESTION 3
What is the vertex of a parabola?
Its turning point — the maximum or minimum point of the curve.

Going deeper? Find the roots.

Have students find where a parabola crosses the x-axis (its roots) by reading the graph or factoring. A printable quadratics sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Quadratic FunctionsFlip
Card 1
Term
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Meaning
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Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Quadratic FunctionsQuiz
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Question 1
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Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Quadratic FunctionsPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: axis of symmetry, coefficient, maximum, minimum, parabola, quadratic function, root, vertex
  1. the highest point
  2. the turning point of a parabola
  3. the line splitting a parabola in half
  4. the lowest point
  5. a number multiplying a term
  6. an x-intercept (where y = 0)
  7. the U-shaped graph of a quadratic
  8. a function with an x² term

Part B · Show what you learned

  1. What shape is the graph of a quadratic function?
  2. In y = ax² + bx + c, what happens to the parabola when a is negative?
  3. What is the vertex of a parabola?
Answer key — Part A: 1) maximum · 2) vertex · 3) axis of symmetry · 4) minimum · 5) coefficient · 6) root · 7) parabola · 8) quadratic function
Part B: 1) A parabola (a U-shape). 2) It opens downward (like an upside-down U). 3) Its turning point — the maximum or minimum point of the curve.