A linear equation in two variables doesn't involve any power higher than one for either variable. It has the general form Ax + By + C = 0, where A, B and C are constants. It's possible to simplify this to y = mx + b, where m = ( −A / B) and b is the value of y when x = 0. A quadratic equation, on the other hand, involves one of the variables raised to the second power. It has the general form y = ax2 + bx + c. Apart from the adding complexity of solving a quadratic equation compared to a linear one, the two equations produce different types of graphs.
TL;DR (Too Long; Didn't Read)
Linear functions are one-to-one while quadratic functions are not. A linear function produces a straight line while a quadratic function produces a parabola. Graphing a linear function is straightforward while graphing a quadratic function is a more complicated, multi-step process.
Characteristics of Linear and Quadratic Equations
A linear equation produces a straight line when you graph it. Each value of x produces one and only one value of y, so the relationship between them is said to be one-to-one. When you graph a quadratic equation, you produce a parabola that begins at a single point, called the vertex, and extends upward or downward in the y direction. The relationship between x and y is not one-to-one because for any given value of y except the y-value of the vertex point, there are two values for x.
Solving and Graphing Linear Equations
Linear equations in standard form (Ax + By + C = 0) are easy to convert to convert to slope intercept form (y = mx +b), and in this form, you can immediately identify the slope of the line, which is m, and the point at which the line crosses the y-axis. You can graph the equation easily, because all you need are two points. For example, suppose you have the linear equation y = 12_x_ + 5. Choose two values for x, say 1 and 4, and you immediately get the values 17 and 53 for y. Plot the two points (1, 17) and (4, 53), draw a line through them, and you're done.
Solving and Graphing Quadratic Equations
You can't solve and graph a quadratic equation quite as simply. You can identify a few general characteristics of the parabola by looking at the equation. For example, the sign in front of the x2 term tells you whether the parabola opens up (positive) or down (negative). Moreover, the coefficient of the x2 term tells you how wide or narrow the parabola is -- large coefficients denote wider parabolas.
You can find the the x-intercepts of the parabola by solving the equation for y = 0 :
ax2 + bx + c = 0
and using the quadratic formula
x = [ −b ± √(b2 − 4_ac_)] ÷ 2_a_
You can find the vertex of a quadratic equation in the form y = ax2 + bx + c by using a formula derived by completing the square to convert the equation into a different form. This formula is −b/2_a_. It gives you the x-value of the intercept, which you can plug into the equation to find the y-value.
Knowing the vertex, the direction in which the parabola opens and the x-intercept points gives you enough of an idea of the appearance of the parabola to draw it.