notes/education/math/Quadratics.md

Notes

To convert an equation in the form of y = (x^2-3x) into a square equivalent, you:

• Take the second value, 3, then half it, giving you (\frac{3}{2}).
• Then to rebalance the equation, you're going to square that value (giving you \frac{9}{4}), then add it to the other side, multiplying it by a if necessary, where a is what the parentheses are multiplied by in the form a(x - h)^2. This will give you an equation that looks something like: \frac{9}{4} + y = (x -\frac{3}{2})^2.
• Finally, you can rebalance the equation by subtracting \frac{9}{4} from both sides, giving you y = (x - \frac{3}{2})^2 - \frac{9}{4}. This equation should be equal to the original.

 y = -5x^2 -20x + 13 

Given the above equation, you can factor out a -5, resulting in the equation -5(x^2+4x) + 13). Half of 4 is 2, and because the inside is multiplied by -5, -5 *4 = -20, so you add -20 to the other side to equalize the equation, resulting in an equation in the form of -20 + y = -5(x+2)^2+ 13. This simplifies down to y = -5(x+2)^2 + 33.

Forms

Standard form (vertex form)

 y = a(x - h)^2 + k 

To convert to standard form given a vertex of a quadratic equation and a point that falls along that line, plug values in for everything and solve for a.

Quadratic form

 y = a^2 + bx + c 

End Behavior of functions

If the largest exponent of a function is even, both sides of a function will point the same way. As x goes towards infinity, f(x) will go towards infinity. As f(x) goes towards -\infty, f(x) will still go to infinity

 x \rightarrow \infty, \space f(x) \rightarrow \infty 

 x \rightarrow -\infty, \space f(x) \rightarrow -\infty