Quadratic vs. linear functions: What's the difference in Algebra 1?

📚 Understanding Linear Functions

A linear function is like a straight path. It shows a constant rate of change, meaning for every step you take forward, you go up (or down) by the same amount. Think of it like walking on a flat ramp!

📈 Definition of a Linear Function

• 📏 Equation Form: Linear functions can be written in the form $y = mx + b$, where:
• ➕ $m$ is the slope (the rate of change, how steep the line is).
• 📍 $b$ is the y-intercept (where the line crosses the y-axis).
• 🔢 Graph: The graph of a linear function is always a straight line.

🤔 Understanding Quadratic Functions

A quadratic function is like a curved path, more specifically a parabola! The rate of change isn't constant; it increases or decreases as you move along the x-axis. Think of it like throwing a ball in the air - it goes up, slows down, reaches a peak, and then comes back down.

🎢 Definition of a Quadratic Function

• 📐 Equation Form: Quadratic functions can be written in the form $y = ax^2 + bx + c$, where:
• ✖️ $a$, $b$, and $c$ are constants.
• 🧮 Key Feature: The $x^2$ term is what makes it quadratic and creates the curve.
• 📊 Graph: The graph of a quadratic function is a parabola, a U-shaped curve.

Quadratic vs. Linear Functions: Key Differences

Feature	Linear Function	Quadratic Function
Equation Form	$y = mx + b$	$y = ax^2 + bx + c$
Graph	Straight Line	Parabola (U-shaped curve)
Rate of Change	Constant	Varying (not constant)
Highest Power of x	1	2
Number of Roots (Solutions)	Usually 1	Up to 2

🚀 Key Takeaways

• 💡 Straight vs. Curve: Linear functions create straight lines, while quadratic functions create curves (parabolas).
• ✖️ The $x^2$ Term: The presence of an $x^2$ term is the biggest indicator of a quadratic function.
• 📉 Constant vs. Varying Change: Linear functions have a constant rate of change (slope), whereas quadratic functions do not.