217. Exploring Linear and Quadratic Relations with Digital Tools

Learning Intentions

• Use digital tools to vary parameters in functions and relations.
• Observe graphical changes caused by parameter variation.
• Record connections between equations and graphs.

Pre-requisite Summary

• Understand linear functions in the form .
• Understand quadratic functions in the form .
• Recognise gradient, intercepts, and basic parabola features.
• Plot points on the Cartesian plane.
• Understand that changing parameters changes graphs.
• Read coordinates and interpret graph features.
• Access GeoGebra for a digital calculation and graphing tool. Otherwise use Desmos

Worked Examples

Worked Example 1

Using a digital graphing tool, compare:

and

Describe what happens when the parameter changes.

Worked Example 2

Using a graphing tool, explore:

and

Describe changes.

Worked Example 3

Compare:

and

Worked Example 4

Explore:

compared with:

Worked Example 5

Using a digital tool, compare:

and

Problems

Problem 1

Use a digital tool to compare:

and

Record what happens to steepness.

Problem 2

Use a graphing tool to compare:

and

Describe the change in shape.

Problem 3

Explore:

and

Record the transformation.

Problem 4

Compare:

and

Describe the shift.

Problem 5

Use a digital tool to compare:

and

Record what changes in the graph.

Exercises

Understanding and Fluency

Exercise 1

Use a digital tool to vary in:

Record what happens as increases.

Exercise 2

Investigate:

and

Describe changes.

Exercise 3

Investigate:

and

Record observations.

Exercise 4

Investigate:

and

Describe the shift.

Exercise 5

Investigate:

and

Describe shape changes.

Exercise 6

Record how changing affects a parabola.

Exercise 7

Record how changing affects curvature.

Exercise 8

Compare linear graphs with different gradients using a digital tool.

Exercise 9

Compare quadratic graphs with different values.

Exercise 10

Match equations to observed graphs using a digital tool.

Reasoning

Exercise 11

Explain why digital tools make it easier to see parameter changes.

Exercise 12

Explain why changing does not affect the shape of a parabola.

Exercise 13

Explain why two different quadratic equations can have the same shape but different positions.

Exercise 14

Describe how you would identify the effect of using a graphing tool.

Problem-solving

Exercise 15

A student explores:

They find the graph gets narrower as increases.

Explain this relationship.

Exercise 16

A graph shows a parabola shifted 6 units up.

Suggest a possible equation and justify your choice.

Exercise 17

Using a digital tool, a student compares:

and

Describe all transformations step by step.

Potential Misunderstandings

• Thinking digital tools automatically explain changes without interpretation.
• Confusing horizontal and vertical shifts.
• Assuming changing only shifts the graph vertically.
• Misinterpreting negative as a vertical shift instead of reflection.
• Believing shape changes when only changes.
• Overlooking that vertex position changes with in expanded form.
• Assuming all graphs must pass through the origin.
• Misreading slider changes as unrelated to parameters.

Next: 218. Graphics Effects from Equations using Digital Tools