1. y = 4 (x – 3)2 – 3
2. = -2 (x – 1) (x + 2)
3. y = 3×2 + 2x + 1

All About Parabolas

In geometrical terms, a parabola is all of the points that are equidistant from a point and a line. A parabola takes the form of a quadratic formula, with an x2 somewhere inside. The x2 of a quadratic formula makes the slope increase or decrease a set amount for each increase in x. This increase or decrease in the slope gives the parabola its curved shape. The function of a parabola can be in different forms, each of which can be useful in different situations. The most common form is…

Standard Form of a Quadratic Function: y = ax2 + bx + c

Standard form is the most simplified form of the quadratic formula. It is useful because it contains the c value, whichis the y-intercept. At the y-intercept x is equal to 0, so everything else in the equation cancels out, leaving y = c. Standard form is also useful because in contains the a value, which determines the shape of the parabola: if it’s a smile or a frown, if it’s thin or wide.

Factored Form of a Quadratic Function: y = a (x – p) (x – q)

The factored form can be found by factoring the quadratic equation. This is done by first factoring out the a value (The same a as in standard form). To find p and q, findtwo numbers that can be added to get the b value and multiplied to get c value of standard form, then multiply them by -1. For example, the quadratic equation y = 2×2 + 6x + 4 would become y = 2 (x + 1) (x + 2). The factored form of an equation is useful because it contains the two x-intercepts of the parabola. At the x-intercept, y is equal to 0, so the equation becomes 0 = a (x – p) (x – q). The Zero Product Principle states that if either one or more of the factors are equal to zero, then the product will also equal zero. This means that x must equal either p or q to make the equation true, because p – p and q – q both equal zero. Therefore, p and q are the x-intercepts. Factored form also contains the y-intercept: it is equal to a • p • q. This works because at the y-intercept x will equal zero, leaving the equation y = a (0 – p) (0 – q). The negatives cancel out, leaving the function as y = a • p • q. The vertex of this equation can be found by finding halfway between the two x-intercepts, because a parabola is symmetrical, then solving for y with that value of x.

Vertex Form of a Quadratic Function: y = a (x – h)2 + k

Vertex form can be extracted from factored form by first solving for the vertex, than placing that into the function so that the x coordinate replaces h and the y coordinate replaces k. Then take the a value from factored form (which is the same a as in factored and standard form) and placing it in the vertex form equasion. The equation y = 2 (x + 1) (x + 2), which has a vertex of (-1.5, -.5), would end up being y = 2 (x + 1.5)2 – .5 in vertex form. Vertex form is useful because it shows what the parabola looks like in a more direct way: a represents the shape of the parabola, h represents how far to the left or right the vertex of the parabola is, and k represents how far up or down the vertex of the parabola is.