Key Differences Between Linear and Quadratic Functions

When we look at linear and quadratic functions, there are some important differences to notice:

1. Basic Forms:

   • Linear Function: It usually looks like this: $y = mx + b$. Here, $m$ tells us how steep the line is, and $b$ is where the line crosses the y-axis.
   • Quadratic Function: This one looks like this: $y = ax^2 + bx + c$. In this case, $a$, $b$, and $c$ are numbers (constants), and $a$ cannot be zero.

2. Shape of the Graph:

   • Linear functions make straight lines. The slope $m$ decides how slanted the line is.
   • Quadratic functions make U-shaped curves called parabolas. The number $a$ shows us which way the U opens. If $a$ is positive, the U-shaped curve goes up. If $a$ is negative, the curve goes down.

3. Degree of the Function:

   • Linear functions are called first degree because the highest power of $x$ is 1.
   • Quadratic functions are second degree since the highest power of $x$ is 2.

4. Solutions:

   • A linear equation can touch the x-axis at most one time, which means it has one solution.
   • A quadratic equation can touch the x-axis up to two times, giving it two solutions. Sometimes, it might not touch the x-axis at all if a specific formula (called the discriminant) gives a negative number.

5. Changing the Functions:

   • We can change linear functions by shifting them left and right or stretching them. Quadratic functions can also be flipped over the x-axis because of the $x^2$ part.

These differences shape how linear and quadratic functions work and how we use them in math.