ACT
equation of a circle
(x - a)² + (y - b)² = r² center at (a, b)
equation of a hyperbola
(x² / a²) - (y² / b²) = 1 OR (y² / a²) - (x² / b²) = 1
integers
-4, -3, -2, -1, 0, 1, 2, 3, 4
digits
0, 1, 2, 3, 4
whole/counting numbers
0, 1, 2, 3, 4...
if an outcome will definitely occur, the probability is...
1
how to solve quadratic inequalities
1) factor 2) plot test intervals on number line 3) test an x-value in each interval and see whichever satisfy the original inequality
how to graph y ≤ 2x
1. graph y = 2x 2. plug in a random point 3. if this makes the inequality true, shade the side of the line containing this point
find the point of intersection of two lines
1. solve for x in first equation 2. plug in this expression for x in the second equation 3. solve for y 4. plug this y into the first equation to form a coordinate (x, y)
degrees in a triangle
180⁰
if given an angle and told to find the arc length...
2πr(x / 360⁰)
Pythagorean triple
3-4-5
what is the sum of the exterior angles of a triangle?
360⁰
can 2√3 + 5√3 be simplified?
7√3
relatively prime numbers
DOES NOT MEAN THEY ARE PRIME NUMBERS it just means the GCF of two numbers is 1
if you have a line segment on a graph, and you want to know its length...
FORGET THE DISTANCE FORMULA create a right triangle with it as the hypotenuse and use Pythagorean Theorem
is 1 a prime number?
NO
should i guess?
YES
midpoint formula
[ (x + x)/2 , (y + y)/2 ]
equation of an ellipse
[ (x - h)² / a² ] + [ (y - k)² / b² ] = 1 (h, k) is vertex → halfway between foci if larger denominator is under x, then the x-axis is the major axis; vice versa with y major axis contains foci and ends in vertices length of horizontal axis is 2a length of vertical axis is 2b
standard form of complex numbers
a + bi
multiplying exponentials with the same base
add exponents
interior angles of an n-sided polygon
add up to 180(n - 2)
interior angles of a quadrilateral
add up to 360⁰
If you know the probability of all other events occurring, you can find the probability of the remaining event by...
adding the known probabilities together and subtracting that sum from 1
3sinx
amp is 3 graph stretched vertically
if given an arc length and told to find the angle...
arc length = (radius) (angle in radians) (180 / pi)
if working with surface area of a rectangular prism, what should you NOT do?
assume that all four non-base faces are equal instead, divide the figure into three groups of three faces, find their surface areas, and add together
difference and sum of cubes
a³ + b³ = (a + b) (a² - ab + b²) a³ - b³ = (a - b) (a² + ab + b²)
area of a parallelogram
base × height
diagonals of a square
bisect each other at right angles
when to change the sign of an inequality
both when you divide AND multiply by a negative when you divide or multiply by a variable, you have no way to know if it was positive or negative, so you have two answers or no solution (I guess)
rational numbers
can be written as fractions, terminating decimals, or repeating decimals
how to find slope when given two coordinates
change in y / change in x
you can only use the fundamental counting principle if...
choices must be independent
multiplication of matrices
columns in first must equal rows in second multiply the first row of the first matrix by the first column of the second matrix add numbers to yield dot product do this for each column of the second matrix then, do ALL this with all the rows in the first matrix
alternate interior angles of parallel lines cut by a transversal
congruent
corresponding angles (on same side of transversal but one is interior and one is exterior) of parallel lines cut by a transversal
congruent
vertical angles (the ones that look at each other) of parallel lines cut by a transversal
congruent
factors
divide evenly into a number 3 is a factor of 12
interior angles of a regular polygon
each alone equals 180(n - 2) / n or just draw all the diagonals from a single vertex and multiply the triangles formed by 180⁰
slope of a line that runs straight through the origin
either 1 or -1
interior angles on the same side of a transversal cutting parallel lines
equal 180⁰
is 0 even or odd?
even
f ₀ g
f(g(x))
system of equations
gives you the point where the graphs of the two equations intersect
how to graph f(x + c)
graph f(x) and shift it c units to the LEFT
how to graph f(x) + c
graph f(x) and shift it up c units
diameter of a rectangle
hypotenuse of two right triangles
similar shapes have...
identical corresponding angles ratios of corresponding sides all equal each other
what are √2 and π?
irrational...numbers never end
how to solve compound inequalities
isolate the variable in the middle by performing the same operations of both ends
how do you evaluate the exterior angle of a triangle?
it is always equal to the sum of the remote interior angles (the angles that are not adjacent to the exterior angle)
how to tell if a number is divisible by 6
it will be even and divisible by 3
how to tell if a number is divisible by two
it will end in a 0, 2, 4, 6, or 8
how to tell if a number is divisible by 5
it will end in a 5 or a 0
how to tell if a number is divisible by 3
its digits will add up to a multiple of 3
how to tell if a number is divisible by 9
its digits will add up to a multiple of 9
how to tell if a number is divisible by 8
its last three digits will be divisible by 8
how to tell if a number is divisible by 4
its last two digits will be a multiple of 4
adding and subtracting 2 × 2 matrices
just add or subtract corresponding parts to create another matrix
scalar multiplication of matrices
just multiply the constant by each number
prime factorization
keep breaking down the number in a tree diagram until all you have left is prime numbers
relationship between angles and side lengths of a triangle
largest angle is opposite longest side and smallest angle is opposite shortest side
how to flip f(x) over the x-axis
multiply by a negative c value so, (-c)f(x)
is 0 positive or negative?
neither
what is the restriction on side length of a triangle?
no side of a triangle can be as long as the sum of the other two side lengths
parallelogram
opposite (across diagonal) angles and opposite sides are congruent adjacent angles are supplementary the diagonals split it into four little triangles (two congruent pairs)
perpendicular bisector of a chord in a circle
passes through the center of the circle
graph of cosx
period 2π
graph of sinx
period 2π
sin2x
period is π b/c (2π / 2) = π graph squished horizontally
graph of tanx
period π
trapezoid
quadrilateral with only one pair of parallel sides
multiples
result of multiplying a number by a number 12 is a multiple of 3
the grade of a path or road
rise/run × 100%
if you know the length of a chord and its distance from the center, and you want the circle's radius...
set up a right triangle with the distance to the center as one leg, half of the chord as the other leg (because the distance to the center must be a perpendicular bisector), and the radius as the hypotenuse
to graph a system of inequalities...
shade the region where the shading of the two separate inequalities would overlap
45⁰-45⁰-90⁰ triangle
side ratios of 1-1-√2
30⁰-60⁰-90⁰ triangle
side ratios of 1-√3-2
trig identities
sin²x + cos²x = 1 tanx = sinx / cosx sin2x = 2 sinx cosx cos2x = cos²x - sin²x tan2x = 2tanx / (1 - tan²x)
if they ask when a system of equations has infinite solutions...
they mean where they are the same line put them in y = mx + b form and look for whatever variable value will make these the same equation
how do you find the diagonals of a rectangular prism?
they stretch from opposite corners, there are four of them, and they are all equal in length diagonal = √l² + w² + h²
how to find intercepts
to find x-intercept, set y = 0 and solve for x to find y-intercept, set x = 0 and solve for y
average speed
total distance / total time
alternate triangle similarity rule?
two triangles are similar if they have two pairs of corresponding angles and one pair of sides that are equal OR if one pair of angles is equal and the two pairs of adjacent sides are proportional
when do you change the inequality symbol?
when multiplying or dividing by a negative
rewrite log ₂ x = 3
x = 2³
If Mark can mow the lawn in 40 minutes and Audrey can mow the lawn in 50 minutes, which equation can be used to determine how long it would take the two of them to mow the lawn together?
x/40 + x/50 = 1 because its rate (1 lawn / 40 minutes or 1 lawn/ 50 minutes) times time (x) = 1 mowed lawn
equation of a parabola
y - k = a(x - h)² opens up or down OR x - h = a(y - k)² opens left or right vertex is (h, k)
indirect variation
y = a / x
direct variation
y = ax
slope-intercept form
y = mx + b b is y-intercept!!!!!!!!
solving systems of equations
you can either do that subtracting thingy OR you can do variable substitution
how to solve absolute value equations
|x + 4| = 9 split it into 2 x + 4 = 9 OR x + 4 = −9
area of a triangle
½ base × height
volume of a cylindar
πr² × height
distance formula
√(x - x)² + (y - y)²
i? i²? i³? i⁴? etc?
√-1 -1 -i 1 pattern repeats every fourth exponent
can √3 × √6 be simplified?
√18
