Modeling Real-World Situations with Quadratic Functions

Introduction

Mission:

You are a structural engineer hired by a private company to design a new skateboard park for your city. They want to build a half-pipe that will accommodate skateboarders of all skill levels. Your job is to research real skateboard park specifications, model the shape of the half-pipe using a quadratic function, and present your findings to the company board. 

Why this Matters:

Quadratic functions are essential tools for engineers and architects. They model parabolic shapes found everywhere, from skateboard ramps and suspension bridges to satellite dishes and water fountains. By understanding how to write, interpret, and manipulate quadratic functions, you'll see how algebra directly impacts real-world design and safety.

Your Challenge:

Can you use mathematics to design a safe, functional skateboard park that meets city specifications and engineering standards? Let's find out!

Task

Your team will complete a comprehensive project with the following deliverables:

Part A: Research & Data Collection

  • Find real skateboard park dimensions and specifications
  • Document depth, width, angle of incline, and safety standards
  • Cite all sources using proper APA format

Part B: Mathematical Model

  • Write a quadratic function representing the half-pipe's cross-section
  • Show all work and explain your reasoning
  • Provide the function in multiple forms (vertex form, standard form, and/or intercept form)

Part C: Analysis of Key Features

  • Identify the vertex, x-intercepts, and axis of symmetry
  • Interpret what each feature means in the context of the skateboard park
  • Explain how these features affect skater safety and experience

Part D: Graphical Representation

  • Create an accurate, detailed graph of your half-pipe model
  • Label all key features with coordinates
  • Use an appropriate scale and clear gridlines

Part E: Design Presentation

  • Prepare a visual presentation (digital slides, poster, or video)
  • Explain your mathematical model to the parks department
  • Address safety considerations and design constraints
  • Justify why your model is the best choice for the city
Process

Step 1: Research Real-World Data (Estimated time: 1–2 class periods)

Your first task is to gather accurate information about skateboard park design standards.

Resources to Explore:

  • Tony Hawk Foundation (tonyhawkfoundation.org) – Professional half-pipe specifications
  • Skate Park Design Standards – Search for documents on typical dimensions and safety requirements
  • YouTube Construction Videos – Time-lapse videos of actual half-pipe construction
  • Local Parks Department – Contact your city or visit local skateboard parks to observe real dimensions
  • Engineering Websites – Search for skateboard park design guidelines and best practices

What to Record:

Create a data table with the following information:

Dimension Your Value Unit Source
Total Width (side to side)   feet  
Maximum Depth (top to bottom)   feet  
Radius of Curvature   feet  
Angle of Transition   degrees  
Coping Height   feet  
Recommended Skill Level   (beginner/intermediate/advanced)  

Important: Document all your sources so you can cite them in APA format later.

Step 2: Set Up Your Coordinate System (Estimated time: 1 class period)

Before you can write your function, you need to establish how the half-pipe will be positioned on a coordinate plane.

Your Coordinate System:

  • x-axis: Horizontal distance across the half-pipe (in feet)
  • y-axis: Vertical height above the bottom of the ramp (in feet)
  • Origin (0, 0): Located at the center bottom of the half-pipe

Example Setup:

If your half-pipe is 40 feet wide and 12 feet deep:

  • Left edge of ramp: (-20, 0)
  • Right edge of ramp: (20, 0)
  • Deepest point (bottom center): (0, -12)

Your Task:

Using your research data, sketch your coordinate system on paper. Mark the key points (edges and bottom) that your parabola will pass through.

Step 3: Write Your Quadratic Function (Estimated time: 1–2 class periods)

Now you'll use your data to create the mathematical model of your half-pipe.

Choose One Form(or all):

Option A – Vertex Form y=a(x−h)²+k

Where:

  • (h, k) = coordinates of the vertex (the deepest point)
  • a = a constant that determines the "width" or "narrowness" of the parabola

Option B – Standard Form y=ax²+bx+c

Option C – Intercept Form (Great if you know the edges!) y=a(x−r_1)(x−r_2)

Where r_1 and r_2 are the x-intercepts (where the ramp meets the ground)

How to Find the Constant a:

  1. Use the vertex form if you know the vertex
  2. Substitute a known point (like one of the edges) into the equation
  3. Solve for a

Worked Example:

Your half-pipe is 40 feet wide with a maximum depth of 12 feet.

  • Edges: x-intercepts at x = -20 and x = 20 (where y = 0)
  • Vertex: at (0, -12)

Using Vertex Form: y=a(x−0)²+(−12); y=ax²−12

Substitute the point (20, 0): 0=a(20)²−12; 0=400a-12; 12=400a; a=.03  

Your Function: y=0.03x²−12

Show all your work! Write out each step so the parks department can follow your reasoning.

Step 4: Analyze Key Features (Estimated time: 1 class period)

For your quadratic function, you must identify and interpret each of these features:

Feature How to Find It What It Means for Your Half-Pipe
Vertex Use x = -b/(2a) or read from vertex form The deepest point; where skaters reach maximum speed
X-intercepts (Zeros) Solve the equation y = 0 The edges where the ramp meets the ground; defines the width
Axis of Symmetry The vertical line x = h (or x = -b/(2a)) The center line of the half-pipe; shows perfect balance
Direction of Opening Check if a > 0 (opens up) or a < 0 (opens down) Should open downward for a ramp; confirms your model is correct
Maximum Depth The y-coordinate of the vertex How deep the ramp is; affects speed and difficulty level

Create a Summary Table:

Feature Your Value Real-World Meaning
Vertex    
X-intercepts    
Axis of Symmetry    
Maximum Depth    
Width at Ground Level    

Safety Considerations:

  • Is your depth within safe limits for beginners? (Typically 6–12 feet)
  • Is your width appropriate for the skill level? (Wider = easier)
  • Do your calculations make sense in the real world?

Step 5: Create a Detailed Graph (Estimated time: 1 class period)

Using graphing technology (Desmos, GeoGebra, graphing calculator, or even graph paper), create a professional graph that shows:

Required Elements:

The parabola clearly drawn (your half-pipe cross-section)

Vertex labeled with exact coordinates

X-intercepts labeled with exact coordinates

Axis of symmetry shown as a dashed vertical line

Grid with appropriate scale (should show the full width and depth)

Labeled axes with units (x = distance in feet, y = height in feet)

Title: "Half-Pipe Cross-Section Model"

Legend if you show multiple functions for comparison

Graph Quality:

  • Use a ruler for straight lines (if hand-drawn)
  • Make sure all labels are clear and easy to read
  • Choose a scale that fills most of the page
  • Use consistent spacing for gridlines

Step 6: Build Your Presentation (Estimated time: 1–2 class periods)

Now you'll create a professional presentation for the parks department. Choose one format:

  • Digital Slides (PowerPoint, Google Slides, Keynote)
  • Poster (Physical or digital)
  • Video (Recorded presentation with visuals)
  • Website or Blog (Interactive presentation)

Required Slides/Sections:

  1. Title Slide

    • Project title
    • Team member names
    • Date

  2. Research Summary

    • Key data you collected (in a table)
    • All sources cited in APA format
    • Photos or diagrams of real skateboard parks (if available)

  3. Mathematical Model

    • Your quadratic function (all forms if possible)
    • Explanation of how you found it
    • Step-by-step work shown

  4. The Graph

    • Your detailed half-pipe graph
    • All features labeled
    • Clear title and axes labels

  5. Feature Analysis

    • Table or infographic showing:
      • Vertex and its meaning
      • X-intercepts and their meaning
      • Maximum depth and width
      • How these relate to skater experience

  6. Safety & Constraints

    • Explain how your model ensures skater safety
    • Discuss any limitations or assumptions
    • Address why your design is better than alternatives

  7. Conclusion Slide

    • Why quadratic functions matter for engineering
    • Real-world applications beyond skateboard parks
    • Key takeaways for the parks department
Evaluation

WebQuest Student Assessment Rubric

Student Name: ________________________ Date: ____________ Team Members: _______________________

Criterion Excellent (4) Proficient (3) Developing (2) Beginning (1) Points
Research & Data (Content) Collected comprehensive, accurate data from 4+ credible sources; all sources clearly documented and cited in APA format Collected accurate data from 3 credible sources; most sources cited in APA format Collected data from 2 sources; some sources cited; minor APA errors Data incomplete or from unreliable sources; citations missing or incorrect ___ / 10
Mathematical Model – Function (Content) Wrote correct quadratic function in all three forms (vertex, standard, intercept); showed complete work with clear reasoning Wrote correct quadratic function in at least one form; showed most work with explanation Function has minor errors; work partially shown; some reasoning unclear Function incorrect or missing; work not shown ___ / 10
Key Features Analysis (Content) Correctly identified and interpreted ALL features (vertex, x-intercepts, axis of symmetry, direction); explained real-world meaning of each Correctly identified most features; explained most real-world meanings clearly Identified some features; limited interpretation; unclear connections to context Few features identified; little or no real-world connection ___ / 10
Step-by-Step Process Clarity (Content) Process described in detailed, sequential steps; each step is clear, logical, and easy to follow; transitions between steps are smooth Process described in clear steps; mostly logical sequence; most transitions are smooth Process described but some steps unclear or out of order; some transitions missing Process vague or disorganized; difficult to follow ___ / 20
Graphing Accuracy & Completeness (Content) Graph is accurate and detailed; all required features labeled with coordinates; appropriate scale; professional appearance; title and axis labels clear Graph is accurate with most features labeled; appropriate scale; clear labels; minor formatting issues Graph has some errors; missing some labels; scale or formatting issues Graph incomplete, inaccurate, or missing key elements ___ / 10
Grading Rubric Clarity (Content) Created clear, detailed rubric for student work; includes specific criteria, performance levels, and point values; easy for students to understand expectations Created rubric with clear criteria and performance levels; most point values appropriate; generally easy to understand Rubric present but some criteria unclear; performance levels not well-defined; point values inconsistent Rubric missing or too vague; does not guide student work ___ / 20
Conclusion Instructions (Content) Provides clear, detailed instructions for conclusion; includes 4+ reflection questions that encourage deep thinking; explains how to wrap up learning Provides clear conclusion instructions with 3 reflection questions; encourages critical thinking Conclusion instructions present but somewhat vague; 1–2 reflection questions; limited depth Conclusion instructions missing or unclear; no reflection questions ___ / 10
Grammar, Spelling & APA Citations (Mechanics) No grammar or spelling errors; all citations in proper APA format; professional writing throughout 1–2 minor errors; most citations in APA format; generally professional writing 3–4 errors; some citations missing or incorrect APA format Multiple errors; many citations missing or improperly formatted ___ / 10
Navigation & Transitions (Mechanics) Smooth transitions between all sections; easy to navigate from one part to the next; clear organization; logical flow Mostly smooth transitions; generally easy to navigate; good organization Some transitions unclear; navigation slightly confusing; organization could be better Transitions missing or confusing; difficult to navigate; poor organization ___ / 10

Total Points: ______ / 100

Grading Scale:

  • 90–100: A (Excellent)
  • 80–89: B (Proficient)
  • 70–79: C (Developing)
  • Below 70: F (Needs Improvement)

Teacher Comments:

Conclusion

After completing all phases of this webquest, you will complete a comprehensive reflection to demonstrate your learning. This conclusion ties together all the mathematics, research, and design work you've done.

Part A: Written Reflection (1–2 pages)

Answer each of the following questions in complete sentences. Show evidence of deep thinking and connection to real-world applications.

Question 1: Connecting Algebra to Real-World Design

How did you use the quadratic function to model the half-pipe? Explain the connection between the algebra (the equation, the graph, the features) and the actual physical shape of the ramp. What would happen if you changed the value of a in your equation? How would that affect the steepness or curvature of the ramp?

Question 2: Interpreting the Vertex

What does the vertex of your parabola tell you about the half-pipe's design? Why is the vertex the most important feature for a skateboard park? How would a deeper or shallower vertex change the skater's experience?

Question 3: What was the most challenging part of this project?

Teacher Page

Grade Level

9–12 (Algebra 2)

Subject Area

Mathematics

Topic/Concept

Quadratic functions, modeling real-world situations, key features of parabolas (vertex, intercepts, axis of symmetry)

Standards Alignment (Florida)

Primary Standards:

  • MA.912.AR.3.4: Write a quadratic function to represent the relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.
  • MA.912.AR.3.6: Given an expression or equation representing a quadratic function, determine the vertex and zeros and interpret them in terms of a real-world context.
  • MA.912.AR.3.7: Given a table, equation or written description of a quadratic function, graph that function, and determine and interpret its key features.
  • MA.912.AR.3.8: Solve and graph mathematical and real-world problems that are modeled with quadratic functions. Interpret key features and determine constraints in terms of the context.
  • MA.912.F.1.8: Determine whether a linear, quadratic or exponential function best models a given real-world situation.