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* = *ax*^{2} + *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 *x*^{2} term tells you whether the parabola opens up (positive) or down (negative). Moreover, the coefficient of the *x*^{2} 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 :

*ax*^{2} + *bx* + *c* = 0

and using the quadratic formula

*x* = [ −*b* ± √(*b*^{2} − 4_ac_)] ÷ 2_a_

You can find the vertex of a quadratic equation in the form *y* = *ax*^{2} + *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.