Difference Between Parabola and Line Equation

Parabolas are the result of graphing quadratic equations.
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When you're graphing equations, each degree of polynomial creates a different sort of graph. Lines and parabolas come from two different polynomial degrees, and looking at the format can quickly tell you what kind of graph you'll end up with.

Linear Equations

Lines come up from first-degree polynomials. The general format for a linear equation is y = mx + b. "M" refers to the slope of the line, which is the rate at which it climbs or falls. A negative slope will go down a graph as x-values decrease, and a positive slope will go up a graph as x-values increase. "B" is called the y-intercept and shows where the line crosses the y-axis.

Plotting a Graph from the Equation

You can plot one point at the y-intercept. So, if you have the equation y = -2x + 5, you can draw a point at 5 on the y axis. Then, plug one more x-value in, such as 3. y = -2(3) + 5 gives you y = -1. So you can draw another point at (3, -1). Draw a line through those points and beyond, drawing arrows on both ends to show the line continues indefinitely.

Parabolic Equations

Parabolas are the result of second-degree polynomials, and the general format is y = ax^2 + bx + c. The "a" indicates the width of the parabola -- the closer l a l (the absolute value of a) is to zero, the wider the arc will be. If "a" is negative, the parabola will open to the bottom; if positive, it will open to the top.

Graphing

You can plug x-values in to find corresponding y-values, but it's trickier to graph because the parabola will curve around a vertex (the point where the parabola turns around). To find the vertex (h,k) divide the opposite of "b" by 2a. In the equation y = 3x^2 - 4x + 5, that gives you 4/3, which is the h-value. Plug h in to get k. y = 3(4/3)^2 - 4(4/3) + 5, or 48/9 - 48/9 + 5, or 5. Your vertex will be at (4/3, 5). Plug in other x-values to get points to help you draw the curving parabola.

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