Quadratics

What to Know

Learning goals

how to graph vertex form

factoring

standard form to factored

first and second difference

what is vertex form and how to graph

an equation, written out in vertex form, is like this: y= a(x-h)^2 +k. in this equation, a, tells us if the parabola will be stretched or compressed, h, tells us where the parabola will be left to right, and k tells us where it would be up or down. When given an equation like 3(x−1)^2​+1, a, which is 3, would tell you that it is stretched by a factor of 3, while h, which is -1, tells you that it is moved 1 unit to the right, since the h sign flips, and k, which is also 1, tells you that the parabola is translated 1 unit up.
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First and second difference

using the first and second difference will help you determine if a graph is linear, quadratic, or neither. to do this, you make a chart, with labels, x and y. write all numbers on x in order from 1 to any number, and for y, you can make a pattern, like 6,3,-3,-12. then multiply each number by the next. if you get a pattern, there is a first difference, indicating that the graph is linear, but if it is not, subtract the first differences the same way to find t he second difference. it is the same, it is quadratic, but if now, it is neither.

Factoring/simple trinomials

there are 2 methods of factoring that are used the most, and that is simple and complex trinomials.

for simple trinomials, it's as simple as finding what multiplies to give you what ever number c is, and what adds up to give you whatever b is. for example, if given an equation like, x^2+8x+16, you have to find what adds to give you 8, but multiplies to get 16, which is 4 and 4. then you put it in brackets like this: (x+ ) (x+ ) and fill it in (x+4) (x+4). since the numbers were the same, you can simplify by doing (x+4)^2.

Factoring/complex trinomials

When solving a complex trinomial, you always want to look for a common factor first. if there is none, continue. what you do next, is multiply a and b to get a product. for example, if given an equation like 6x^2+7x-3, you multiply 6 and -3 to get a product, which is -18. then, you want to find 2 numbers that multiply to give you the product, but add up to give you b, which in this case, is 7. those numbers are 9 and -2.

now, rewrite the equation, replacing b with the 2 numbers like this : 6x^2 +9x-2x-3. now there are 4 variables. split them into to, giving you [6x^2+9x] [-2x-3]. now you find the individual factors of what's in those brackets, giving you: 3x{2x+3} and -1{2x+3}. the last step is to take whatever is in the brackets as your first factored equation, and then, the 2 numbers outside the brackets as your second, like this: {2x+3} {3x-1}

Factoring Complex Trinomials.mp4