Mathswell MATHSWELL
MathPal - Your AI Study Buddy
Learns alongside you • Can make mistakes • May be occasionally unavailable

Functions: Linear vs Quadratic

Discover how straight lines and curves behave differently in mathematics

📏 Linear Functions

Create straight lines that never curve

y = mx + b

• Same change every step
• Constant slope everywhere
• Graph is always straight

📈 Quadratic Functions

Form U-shaped curves called parabolas

y = ax² + bx + c

• Rate of change varies
• Has a turning point
• Creates symmetric curves

🎮 Quick Interactive Demo

Click a preset to see the difference:

Scale: Each grid square = 2 units
Function: y = x
Type: Linear
Key Feature: Constant rate of change

💡 Key Insights

Two Ways to Think About Functions:

1. As a Machine: Put in x, get out y. Linear functions add/multiply by fixed amounts. Quadratic functions square the input.

2. As a Pattern: Linear patterns grow steadily. Quadratic patterns accelerate - they grow slowly then quickly (or vice versa).

Function Explorer

Adjust the controls to see how functions change

📏 Linear Controls

1.0
0.0
Equation:
y = 1.0x + 0.0
Scale: Each grid square = 2 units

Challenge with MathPal

Match your function to the mystery graph!

📝 Mystery Function

Can you recreate this function? Adjust the sliders below to match the green graph!

Scale: Each grid square = 2 units

🎯 Your Function (Blue)

Start with y = x and transform it to match:

1.0
0.0
Your equation: y = 1.0x + 0.0

🤖 MathPal's Thinking

Let's figure this out together! I'll look at the graph and see what patterns I notice. Try adjusting the sliders and see how close you can get!