discriminant = 0
1 solution
discriminant
b2 minus 4 ac
in vertex formula, h =
b/2a
in vertex formula k =
c - b^2/4a
discriminant is negative
no solution
discriminant is positive
2 solutions
standard form shows
where parabola opens; c = y int
positive a in parabolic formulas
parabola opens up
negative a in parabolic formulas
parabola opens down
factored form shows
x intercepts
vertex form shows
vertex
standard formula
ax^2 + bx + c
vertex formula
a(x-h)^2 + k, where (h, k) are the vertex coordinates
factored formula
a(x-b)(x-c)
f(x-c) shifts the function
right c units
f(x+c) shifts the function
left c units
f(x) + c shifts the function
up c units
f(x) - c shifts the function
down c units
-f(x) causes a change in…
y-value
f(-x) causes a change in…
x-value
linear equation
y=mx+b
30-60-90 triangle side rules
x, xroot3, 2x
45-45-90 triangle side rules
x, x, xroot2
distance formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Quadratic Formula
x = -b ± √(b² - 4ac)/2a
Sum of Perfect Squares Addition (a+b)(a+b)
(a^2 + 2ab +b^2)
Sum of Perfect Squares Subtraction (a-b)(a-b)
a^2 -2ab +b^2
difference of perfect square (x2-y2)
(x+ y) (x-y)
x^22/x^10
x^12 (sutbract exponents when dividing w/ same base)
x^3 * x^4
x^7 (sum exponents when multiplying with same base)
(x^3)^4
x^12 (multiply exponents when one is outside the parentheses)
x^-n NEGATIVE EXP RULE
1/x^n (negative exponent is same as 1/x to that exponent’s abs value)
x^(2/3)
cube root of x^2 (numerator is power x is raised to, denominator is root)
every number over 1/3
cube root rulez
compound interest formula
current balance = principal * (1 + (interest rate/number of times compounded)^(time)(# times copounded)
compound interest
Interest paid on interest previously earned; credited daily, monthly, quarterly or semiannually
interest
A sum paid or charged for the use of money or for borrowing money
simple interest formula
SI = P × R × T, where P = Principal, R = Rate of Interest, and T = Time period
right trig?
sohcahtoa
circle formula
(x-h)^2 + (y-k)^2 = r^2
circle area
A=πr²
Circle Circumference
C=2πr
radians to degrees conversion
radians x 180/π
degrees to radians
multiply by pi/180
same slope, different y-intercept
parallel lines/ no solution
Same slope, same y-intercept
infinite solutions same line
Different slope, different y-intercept
one solution
flip over x axis
(x,y) -> (x,-y)
flip over y axis
(x,y) -> (-x,y) (reflection)
volume of a cylinder
V = πr2h
geometry?
1- pithagoras- r triangles 2- sohcatoa - r triangles 3- circles: remember formulas and rules, apply x/2x strat equalling to total same with any shape^^
every linear equation is
a modification of y=x
PERCENT CHANGE
old-new/old x 100
geometry strat 2 congruence
check for what the problem is telling u: equilateral, isosceles
similar triangles
same ratios! --corresponding angles,etc
congruent triangles
2 triangles are congruent if and only if all pairs of corresponding sides and angles are congruent / equals. rotate and match to find angle
inequality solution?
dot inside both shaded regions (desmos)
equal equations
all terms===
ax2 plus bx plus c sum of solution???
-b/a
ax2 plus bx plus c multiplication of solutions??
c/a
no solution in standard form
same m dif b // A/D=B/E=// C/F
infinite solutions in standarf form
same everything A/D=B/E
1 solution in standard form
dif m's A/Ddoesnt= B/E
roots?
square every term
squares?
root every term
f(-9)=f(3)
parabolas are symmetrical// vertex is in middle of 2 numbers //h=-3 beware of signs
parabolic wp?
-b/2a or discriminant
inequality?
variable on the left== flip sign
estimates solution?
wrong usually; redo
ez problem
slow down to get #'s right
divide or multiply by a negative number in inequality
flip signs
at least in inequality
same or greater than symbol
proportion setup
part of whole times total
similar triangles
Two pairs of corresponding angles are equal. : Three pairs of corresponding sides are proportional. : Two pairs of corresponding sides are proportional and the corresponding angles between them are equal.
x inequality
always on left side of equation, if switched = swap signn means less than or greater than
no less than c
≥c
"No more than c"
≤c
Less than or equal to c or "at most c"
≤c
greater than or equal to or "at least c"
≥c
system of linear ineq WP
set up an annotate (SIMPLY AND UNDERSTAND) totals signs- equations signs totals!!-- annotate which units are beingg x2 identify well!! LOCATE/CREATE IS** solve if necessary
usually but not always systems of inequalities will solve for two units:
money amount *usually identified when given 2 or even more totals!
beware if WP is asking for total distance or average distance
apply formulas accordingly eg sum/amt
distance
speed x time set proportions accordingly
syst of linear inequalites
plug in values in equation till u reach max based of data
max or minimum value in a WP?
SOLVE FOR 1 PLUG IN EG X= (Y-2) use results and plug in maximixing one value and minimizing the other or vice versa
rate
A comparison of two quantities by division part to part part to whole
proportion
comparison of 2 ratios usually for an unknown value
ratio
result of the division of a ratio 15m/3s = 5m/s!! it combines 2 units
unit price
total price/ units
speed
distance/time
ratio problems
find ratios unit conversion total? estequiometria set up lin equations sometimes careful of square units
linear function problems
plug in to make it 0!!
formula of area of circular sector
A=21θr2 in radians no degrees
formula of circular arc
s=rθ in radians ofc!
when a problem is given in 0.2
the answers might be divided by 100 dont get confused what do these numbers mean?? leave those time consuming questions for end
break even point
solutions of a factored form equation
common mistake
not 2x checking for a few secs leaving time consuming questions for beginning thinking and mentalizing there is not enough time== there is== not checking all at the end and just running out of time
solution: do mocks and mini sections ! ! ! !
units
usally complex units are ratios instead// total unitsx average makes no sense
proportion in wp pattern
total/unit times measurement
circular proportion
central angle/circular angle= sector area/circular area orrr circumference (ratio of 1 = ratio of other, ratio of other=) also central angle is same as its arc