Chapter 2 Flashcards
Finding y-intercept in a parabola
when x = 0
Finding x-intercept in a parabola
when y = 0 (there can be multiple)
Finding axis of symmetry in a parabola
x in the vertex or -b/2a or h
Finding vertex in a parabola
to find x: axis of sym
to find y: f(aos)
(x, y) or (h, k)
standard -> vertex
complete the square
complete the square
y = x2 - 4x - 5
- set equation to zero: 0 = x2 - 4x - 5
- move the constant term: 5 = x2 - 4x
- take half of the coefficient of x and square it: 4x->2->4
- add to both sides: x2 - 4x + 4 = 5 + 4
- rewrite left side as squared: (x - 2)x2 = 9
- take the square root of both sides: x - 2 = ± 3
- isolate x: x = ±3 + 2
completing the square if x2 has a coefficient
y = 4x2+20x+25
- set equal to zero: 0 = 4x2+20x+25
- divide the polynomial by 4: x2 + 5x + 25/4 = 0
- move the constant term: x2 + 5x = -25/4
- take half of the coefficient of x and square it: 5->5/2->25/4
- add to both sides: x2 + 5x + 25/4= -25/4 + 25/4 -> x2 + 5x + 25/4 = 0
- rewrite left side as squared: (x + 5/2)2 = 0
- take the square root of both sides: x + 5/2 = 0
- isolate x: x = -5/2
vertex -> standard
y = 2(x - 3)2 + 4
- expand the squared binomial: 2(x-3)(x-3) + 4
- foil: 2(x2 -6x + 9) + 4
- distribute: 2x2 - 12x + 18 + 4
- combine: 2x2 - 12x + 22
Long division
fill in the non x’s with 0
Synthetic division
fill in the non x’s with 0
two arrows pointing the same direction
even
two arrows pointing in opposite directions
odd
even pointing up
positive x
even pointing down
negative x
odd starting down ending up
positive x
odd starting up ending down
negative x
if a line goes straight through x-axis
1 root
if a line just touches then bounces off the x-axis
2 roots
if a line goes through the x-axis but holds on the x-axis longer
3 roots
i
√-1
i2
-1
i3
-i or -√-1
i4
1
how to check what i to a high power is
i735
- divide the power by 4: 183.75
- multiple the whole number by 4: 732
- subtract to find remainder: 3
- remainder = power: i3 or -i
Note: if the remainder is 0 it means to the power of 4
How to solve complex numbers
if the denominator has a complex number multiply both the top and bottom by the reciprocal of the bottom. then solve.
Finding roots
use synthetic division (L/F)
Last/First
Factors of the last number / Factors of the first number
Miracle formula
x2 - sumx + productx
Solving rational
FACTOR FIRST
Finding y-intercept in a rational
set x equal to zero
Finding x-intercept in a rational
set the numerator equal to zero (can be multiple)
Finding vertical asymptote in a rational
set the denominator equal to zero (can be multiple)
Finding horizontal/slant asymptote in a rational
- Look at the highest powered x on the numerator and denominator
- If the denominator’s x is to a higher power: y = 0
- if the numerator and denominators x is to the same power: numerator x’s coefficent / denominator x’s coeffiecent
- if the numerator’s x is to a higher power: there is no horizontal asymptote, but a slant or oblique asymptote
Finding slant asymptote in a rational
- Divide the num by the dem using long division
- Ignore the remainder
- y = the answer (ignoring rem)
Hole in rational
after factoring, if any binomials in the denominator and numerator are the same = hole
