act math

area of a trapezoid

(average of bases)(height)

percent change

(change / old) * 100

to find the sum of interior angles in a polygon

(n-2)180

(x/y)^m

(x^m / y^m)

if x+a is a factor of an equation

-a if plugged into equation will equal 0

sum of the roots for P(x)=ax²+bx+c=0

-b/a

45-45-90 triangle

1,1,√2

30-60-90 triangle

1,√3,2

discriminant values

1. discriminant= 0, one solution 2. discriminant= positive, two solutions 3. discriminant= negative, no solutions

completing the square

1. set equation equal to 0 2. set c apart from equation 3. add (b/2)² to both sides 4. factor the equation as (x-a)² and add c

congruent angles

1. vertical angles 2. alternate interior angles 3. corresponding angles (angles that are the same on a different set of angles below)

a lightbulb manufacturer is delivering an order of 6000 lightbulbs using large boxes and small boxes. A large-size box can fit a maximum of 800 lightbulbs. A small-size box can fit a maximum of 350 lightbulbs. If the manufacturer will use 3 large-size boxes, what is the minimum number of small-size boxes needed to deliver all the lightbulbs?

3(800) + 350x = 6000 350x = 3600 x= 10.28... therefore, x has to be greater than 10.28- or 11

(4x)³ =

4*4*4 = 64 x*x*x = x³ an equation raised to an exponent is the ONLY time an exponent is multiplied if the equation were however: (4x)x³, this would be addition of exponents as there is an x being multiplied

an apartment building has only 1 bedroom and 2 bedroom. A construction company was asked to renovate all the apartments to modernize their design. Renovation costs averaged 3200 for each one-bedroom apartment and 4000 for each two-bedroom apartment. if the ratio of one bedroom to two bedroom apartments in the building is 5:3, what was the average renovation cost?

5 one-bedroom apartment cost: $16000 3 two-bedroom apartment cost: $12000 5(one) + 3(two) = $28000 / 8 = $3500

in an equilateral triangle, the angle measures are all

60

for a given show, a theater charges $80 for a premium ticket and $30 for a regular ticket. If a total of n tickets were sold and p of them were premium tickets, what is the total amount, in dollars, from the sale of regular tickets and premium tickets?

80p + 30r p + r = n r= n-p 80p + 30(n-p) 80p + 30n - 30p 50p + 30n

Tom drives 30 miles at an average rate of 50 miles per hour. If Leona drives at an average rate of 40 miles per hour, how many more minutes will it take her to travel the same distance?

DiRT distance = rate * time 30 = (50)(time) time = 3/5 time = 36 minutes (30) = (40) (time) time = 3/4 time = 45 minutes

Quadratic Formula

[-b ± √(b² - 4ac)] / (2a)

-f(x)

across x axis

f(-x)

across y axis

moving left means

adding to x

integers

all positive and negative numbers whose decimal point ends in .0

real

all rational and irrational numbers

in right triangles

a²+b² = c²

absolute value functions can never go

below the x axis

least common multiple

between 2 numbers, the least common multiple is the lowest multiple that two numbers have in common

discriminant

b²-4ac

product of the roots for P(x)=ax²+bx+c=0

c/a

rational numbers

can be expressed as fractions or integers

rotating a point 90 degrees means

changing the x and y coordinates and adding a negative sign depending on which axis was reflected across

conjugate

conjugate of a-bi is a+bi the conjugate of a complex number multiplied by the complex number is the ONLY time a complex product will equal a real number

dirt

d= rate * time

to find the horizontal asymptote

divide leading terms of both numerator and denominator as long as they are they same degree

domain is to ___ range is to ____

domain: x range : y

if a point is rotated about the origin

draw a line from that point to the origin and construct the angle

whenever there is a fraction within a fraction

either push the numerator down or multiply the bottom most with the numerator

f(x) = x²-1 g(x) = 3x+4 what is f(g(x))

f(3x+4) = (3x+4)² - 1

Pierre rents a phone to use during his stay for 3 days. he has budgeted $40 for phone rental. Fast lines charges $8 per day and $0.4 per minute. Talk europe charges $5 per day and $0.5 per minute. which company allows Pierre to talk more, and how many minutes more?

fast: 24+ 0.4x = 40 x= 40 minutes talk: 15+ 0.5x = 40 x= 50 minutes talk offers 10 more minutes of talk time

for what values of x does f(x)=0

find points that touch the x-axis

in a table of values, what is inverse of f(3)

find the x where the output of f(x)=3

similar triangles

have ratios of similar sides that are the same the perimeters of similar triangles also have the same ratio

the degree of an equation states

how many times that equation can cross the x axis

i multiplied

i = i i² = -1 i³ = -i i⁴ = 1 i⁵ = i i⁶ = -1 i⁷ = -i i⁸ = 1 use i⁴ to simplify higher powers

adding complex numbers

i is just a variable (3+5i) - (2-3i) = 1+8i

exterior angle theorem

in a triangle with angles a,b,c if x is on a straight line adjacent to c (x=180-c) then x=a+b note that this theorem only applies when the exterior is on a straight line

the number of workers it takes to build a house varies inversely with the number of days it takes to build that house. If 20 workers are needed to build a house in 15 days, how many workers are needed to build a house in 50 days?

in inverse relationships, xy = k (20)(15) = k 300 = k 300 / 50 = 6

moving left/right affects the _ coordinate moving up/down affects the _ coordinate

left/right = x coordinate up/down= y coordinate

moving a function left or right

means adding inside parentheses and squaring only the parenthetical portion x²-3 moved 2 units left would be: (x+2)² -3

moving a function up or down

means adding or subtracting outside the parentheses f(x) = (x-2)²-2 is moved two units down

complex fractions

multiply both numerator and denominator of the fraction by the conjugate of the denominator

number of options

multiply the number of options in the first choice times the number of options in the second choice

primes

numbers that are only divisible by 1 and itself

perpendicular slopes are

opposite reciprocols

when an option choice includes x and |x| for all real numbers x, |-x| + |x| = a) 2x b) |2x|

plug in both x and -x into the options say x = -3 3 + 3 = 6 = |2x| 2x = 2(-3) = -6

distance =

rate * time

inverse of a function

replace x and y get the equation to equal y

supplementary angles

same side interior

for what value of x is the function f undefined, for what x value is there no y value?

set denominator equal to 0

to find the vertical asymptote

set denominator equal to 0

what are the possible values of x such that xy² = 48 where x and y are integers A) 2,3 B) 2,4 C) 1,12,48 D) 3,12,48 E) 1,3,12,48

since x and y are integers, 48/x should equal a perfect square

moving right means

subtracting from x

adding complex radicals

take out the i and add the complex numbers on calculator √-2 + √-8 = i√2 + i√8 = 3i√2

vertical asymptotes occur when

the denominator can be set to zero and a x value exists

when exponents are being divided and the denominator is larger than the numerator x³/x⁶ =

the denominator is being subtracted from numerator, so the result is x^(-3), this can also be written as 1/x³

when exponents are being divided and the numerator is larger than denominator x⁶/x³ =

the denominator is being subtracted from the numerator x^(6-3) = x³

horizontal asymptotes occur when

the leading terms have the same degree and can be divided in a fraction

least common denominator

the least common denominator is the equation between fractions that has components of both denominators when the two denominators have nothing in common, the lcd is the product of both denominators when the two denominators have components in common, the lcd contains that component only to the power of 1 1/(x²-1) + 1/(3x+3) what is the lcd? x²-1 is (x+1)(x-1) 3x+3 is 3(x+1) the lcd is the product of these two, but since the (x+1) is common it stays to the power of 1 the lcd is 3(x+1)(x-1)

the largest angle in a triangle is always opposite

the longest side

multiple of a number

the multiples of 5 are 5,10,15,20,25

if two lines are perpendicular

the product of their slopes is -1

domain

the set of all possible input values (x) to a function

range

the set of all possible output values (y) to a function

the smallest angle in a triangle is always opposite

the smallest side

the vertex is the midpoint of

the two roots

in an isosceles triangle,

the two sides equal in length have equivalent angles opposite them

in a table of x and f(x) values, to find f(-1)

the x value is -1 find the f(x) corresponding to that value

complex solutions to equations

there are always two solutions to complex equations x= bi and x= -bi (x-2)² = -16 x-2 = 4i x= 4i + 2 (x-2)² = -16 x-2 = -4i x= -4i + 2

absolute value of equation if x<-2, then |x+2| =

there are two parts to the absolute value of an equation 1. the equation as it is inside absolute value sign 2. a negative sign placed in the front of an equation this equation will become either x+2 or -x-2 since x<-2, plug in a number greater than positive 2 into each, say x=3 since |3+2| = x+2 and |3+2| does not = -x-2, the correct answer is x+2

there are 210 students at Salem High School, where the ratio of juniors to seniors is 3:4. There are 140 students at Swampscott High School, where the ratio 4:3. The two schools hold a sports competition and all the juniors and seniors from both schools attend. What fraction of the students at the competition are seniors?

think about basic math, suppose there are 30 juniors and 40 seniors (making the ratio 3/4), this means there are 70 students. since the number is tripled, there are 90 juniors and 120 seniors for a total of 210 students at salem. suppose there are 40 juniors and 30 seniors (making the ratio 4/3), this means there are 70 students. since this number is doubled, there are 80 juniors and 60 seniors, and a total of 140 students at swampscott. (60+120) / (140+210) = 18/35

complex products

to find products, distribute and use i² = -1 as a benchmark (4+i)(5-2i) = 20 - 8i + 5i - 2(-1) = 22-3i

how many imaginary solutions exist to equations

two

a square root equation always has

two solutions

radical equations such as x²-5 = 0 ALWAYS have

two solutions, √5 and -√5

if there is a fraction in the denominator like (1/(2/3))

when there is a fraction in the denominator, a/(b/c) flip the denominator so it becomes ac/b take what is the at the bottommost and multiply it to the numerator

if there is a fraction in the numerator like (1/2)/3

when there is a fraction in the numerator, (a/b)/c, push b down and multiply with c so it becomes a/bc

x is to ____ y is to ____

when there is a vertical asymptote, one value of x does not exist when there is a horizontal asymptote, one value of y does not exist

(x^m)/(x^n) =

when two bases are divided, subtract the exponents x^(m-n)

x⁹ * xⁿ =

when two bases are multiplied, add the exponents = x^(9+n)

vertical asymptotes start with

x

x² + 9 = 0 what are the two complex solutions

x = 3i x = -3i

intersection between equation and x axis

x-intercepts are the number of factors in an equation (x-3)(x-4)(x+5) intersects the x axis 3 times (x-3)(x-4)(x-4) intersects the x axis twice

inversely proportional

xy = k in order to solve an inverse problem, find k first

f(x) is the same as

y

horizontal asymptotes start with

y

reflecting across y axis: changes _ coordinate reflecting across x axis: changes _ coordinate

y axis changes x coordinate x axis changes y coordinate

vertex form

y= a(x-h)² + k

let f(x) = bx - 8 the line passes through (2,6), what is the value of b?

y= f(x) 6 = 2x - 8 14 = 2x 7 = x

directly proportional

y= kx where k is constant

distance formula

√(x2-x1)² + (y2-y1)²

what is √72 + √18

√72 + √18 DOES NOT EQUAL √90 √72 = 6√2 √18 = 3√2 3√2 + 6√2 = 9√2