math (1201) - Quad: + and x of roots

from now on im going to use * instead of x for multiplication, should be easier to understand without mixing it up with x-axis and all that crap, computer nerds would understand.. these are my notes anyway.

so the proof of this comes frooooommm..

ax^2 + bx + c = 0

what if we dont want the a there?

then its:

x^2 + (b/a)x + c/a = 0

coz that a is always annoying to work with, at least if its not 1

the deal is, we know the roots, lets call em A and B, multiply to get c and add to achieve b

so.. if you wanna get the sum of A and B, it'd be equal to the b value over the a value, aka -b/a, catch it?

i forget why its minus though, but oh wellers, i missed a fraction of the lesson

and to find the product, its c/a

math (1127) - Quad: quadratic identities

pretty straight forward, if two expressions are equivalent, all the corresponding coefficients must be equal.

you can use this to solve retarded versions of the quadratic function y = ax^2 + bx + c

i dont know what i can say.. its so weird

math (1126) - Quad: discriminant

got rather scared today when i found out our marks will always be compared to the rest of the grade from now on.

eep -.-

and theres so many more 3u people than us.

gay.

anyway.. besides seeing the proof of how the quadratic formula came to be, we worked with just one simple rule with discriminants.

let y = ax^2 + bx + c

if /\ < 0, y has no real roots, aka doenst collide witht he x-axis

now in terms of taht

if a > 0, and /\ <> 0 for all x values (aka the graph is concave up and above the x-axis)

if a < 0, and /\ < 0, the function is negative definite, and y < 0 for all x values (aka the graph is concave down and below the x-axis

math (1125) - Quad: quadratic inequalities

went over a few examples of quadratic inequalities today.

we highlight different parts depending on the sign

lets say

y = ax^2 + bx + c

if y = 0, highlights are on x-axis

if y > 0, highlights are above x-axis (arrows)

if y < 0, highlights are below x-axis (vertex)

concavity will change the graph, keep that in mind.

ooh i see how it goes. for negative numbers, y <> 0 is highlighted on the vertex.. so its still above/below

ok thats awesome.

notes finished for tonight ^^

math (1124) - Quad: roots in graphs

the discriminant (/\ - err.. this is meant to be at riangle), as we know is taken from the quadratic formula (its the bit under the root sign)

/\ = b^2 - 4ac

if /\ < 0
there will be no solution for ax^2 + bx + c = 0 so that means there will be no x-intercepts

if /\ = 0
there will only be one solution, and the graph will touch the x-axis once (at its vertex)

if /\ > 0
the graph would go through x-axis, intercepting at two points

math (1120) - Quad: y=a(x-u)^2

y=a(x+u)^2 or y=a(u-x)^2

with equations that look like this, we discovered thaat:

- a determines the concavity (if its negative it will point down)
- a also determines the steepness (wideness)
- u determines teh vertex (-u, 0)
solve bx + u = 0 for vertex (-u/b, 0)

math (1116) - Quad: quadratic polynomial

new topic started today: the quadratic polynomial

its pretty straight forward, today we just went through dot points and the basics, revision from all that parabola stuff we did earlier in the year.

man i did not miss them -.-

take forever to do.. so much space

anyway.. terms we have to get to know and use properly:

monic - quadratic where the value before x^2 is 1
coeffcient - the number before the pronumeral basically
leading term - the term with the highest power (x^2)
constant term - the one with no pronumeral
root - x co-ordinates where y = 0 (ie. the x-intercept)
zero - the same thing as root

we all know the quadratic funcion:

y = ax^2 + bx + c

got it memorized?

and if the coeffcient of x^2 is positive, the graph will be concave up, if negative it'll be concave down.

note that as the coeffcient of x^2 gets larger, the graph becomes narrower, and as the coefficient becomes smaller, the graph becomes wider

all basic stuff really.