Algebra II Lesson
Parabolas in Flight: Transforming Quadratics Through Projectile Motion
Shared by A STEM Educator
10th & 11th Level II3-act-mathCCSS.MATH.HSF.BF.B.360 min
Learning Objective
Students will be able to identify and describe the effects of parameters a, h, and k on the graph of f(x) = a(x − h)² + k (CCSS.MATH.HSF.BF.B.3), connecting each transformation to a real-world projectile or arch context.
Lesson Overview
Students explore how the vertex form f(x) = a(x − h)² + k models real-world parabolic arches and projectile paths by discovering how parameters a, h, and k transform the parent parabola y = x².
Materials
- Desmos graphing calculator (student devices or projected)
- Printed task sheets / graphic organizer
- Colored pencils (for before/after graph comparisons)
- Whiteboard markers
- Ruler
Scaffolded Task Progression
1Task 1
Look at the football's parabolic arc in the image. Sketch what you think the path looks like on a coordinate grid where the x-axis is horizontal distance (yards) and the y-axis is height (feet). Mark where you think the peak (vertex) is. Then complete the estimation table:
| Your Estimate | |
|---|---|
| Too Low peak height | _____ ft |
| Too High peak height | _____ ft |
| Best Guess peak height | _____ ft |
| Approximate x-location of peak | _____ yds |
2Task 2
The parent parabola is y = x². Use the interactive Desmos graph below to explore how changing a, h, and k moves and reshapes the parabola. Drag the sliders and record your observations in the table.
[ Interactive graph / diagram — available in the full lesson ]
Observation Table — fill in what you notice:
[ Interactive graph / diagram — available in the full lesson ]
Observation Table — fill in what you notice:
| Change This | What Happens to the Parabola |
|---|---|
| Increase k (k > 0) | |
| Decrease k (k < 0) | |
| Increase h (h > 0) | |
| Decrease h (h < 0) | |
| Make a > 1 | |
| Make 0 < a < 1 | |
| Make a negative (a < 0) | |
| Make a = 0 |
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