ACT Math
Graphs of Sine and Cosine
y=asinBx -amplitude: how tall the graph is -period: 2π/B
three types of quadratic equations
y=ax^2+bx+c: standard: tells y-int y=a(x-p)(x-r): intercept form: tells x-int y=a(x-h)^2+k: vertex form: tells vertex
Examples: A company has to select 3 officers from a pool of 6 candidates. How many different ways can this be done if: (a) The officers are distinct? (b) The officers are not distinct?
(a)P(6,3) = 6! /(6-3)! = 6·5·4 = 120 ***use math, then PRB, then #2 (b)If the officers are not distinct, the triples (s1, s2, s3), (s1, s3, s2), (s3, s2, s1), etc. are the same since the positions are the same. C(6,3)= 6! /((6-3)!3!) =6!/3!3! =20
Equation for ellipse
(h,k) is center, 2a is horizontal axis (width), 2b is vertical axis (height).
Equation for Circle
(h,k) is center, r is radius. a circle centered at the origin has the equation x^2+y^2=r^2
Arc Length of a Sector (Circle)
(n/360)(2πr), where n is the central angle.
Area of a Sector (Circle)
(n/360)(πr²), where n is the central angle.
Midpoint Formula
(x₁+x₂)/2, (y₁+y₂)/2
Polygons
- Sum of angles in a n-sided polygon: (n-2)180 - Angle measure of each angle in a regular n-sided polygon: (n-2)180/n
Ratio for 45:45:90 Triangle/Isosceles Right Triangle
- x stands for the side of the figure - This applies to an isosceles triangle Ex: If a question tells you that the side of a square is 5 and wants to know the diagonal of the square, you know immediately that is 5 square root of 2
Ratio for 30:60:90 Triangle
- x stands for the side of the figure - Whatever the value of the short side of a 30-60-90 triangle, the hypotenuse is always twice as large. The medium side is always equal to the short side times the square root of 3.
Vectors
-start at origin -make up components -add/sub
Working with advanced systems of equations
-try to solve or PLUG IN!
Distance between parallel lines y=2x+4 and y=2x−1?
1. find the perpendicular slope (opposite reciprocal) 2. find a point on one of the lines, and then using the perpendicular slope, find point on second line 3. use distance formula
Solving systems of linear Equations
1. substitution 2. elimination 3. back solving (use answers to solve problem) 4. Graphing (plug both equations in the calculator and see where they intersect)
Area of a triangle without a height
1/2(xysinϑ) **xy are adjacent sides to ϑ
there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 3 of them:
10x10x10=1000 permutations
Distinct Permutations of MISSISSIPPI
11!/{(4!)(4!)(2!)}
***Combinations if repetition is not possible
26x25x24x10x9x8
Combinations with 26 letters and 10 digits ***3 letters and 3 digits ***if repetition is possible
26x26x26x10x10x10
The formula for the circumference is
2πr or πd
What is the slope of the straight line passing through the points (-2,5) and (6,4)?
4-5/6- -2= -1/8, this graph will rise 1, and go left 8.
Permutations 5 pens, dif colors
5x4x3x2x1
Distinct Permutations of PEOPLE
6!/[(2!)(2!)]
Trapezoid
A four-sided figure in which two sides are parallel. - Area is 1/2h(base 1+base 2) - You can divide it by two triangles, or two triangles and a rectangle/square
Horizontal Asymptotes
A graph has a horizontal asymptote when, in its most simplified form, it has a numerator and denominator with the same degree (AKA the same highest power of the variable, y=(3x^2+x+10)/(x^2+6x+4) will have a horizontal asymptote because the highest power of x is 2 on both the top and the bottom of the fraction. To calculate where the asymptote will bed divide the leading coefficient of the numerator by the leading coefficient of the denominator. The horizontal asymptote of the function will be at y=3.
Vertical Asymptotes
A graph has a vertical asymptote when, in its most simplified form, a certain value of x will create a zero in the function's denominator. For example y=(x^2+3x+2)/(x-5) will have a vertical asymptote at x=5 because when x=5, the denominator is 0.
Transversal
A line that intersects two or more lines
Rational number
A number that can be expressed as the quotient of two integers and has a finite number of decimal places, or has decimal values that repeat in a pattern.
Irrational number
A number that cannot be expressed as the quotient of two integers and has an infinite number of decimal places. Examples: 3.1415926535897932348..., e (2.7182818284590...), etc.
Rhombus
A parallelogram with 4 congruent sides. - You can divide it by two triangles, or two triangles and a rectangle/square
Parallelogram
A quadrilateral with two pairs of parallel sides. - Opposite angles are equal - Opposite sides are parallel and equal A=bh
Circle
A round shape that has no beginning or end.
Chord
A segment whose endpoints are on a circle
compound interest equation
A=Amount P=principal (aka initial investment) r=interest rate n=number of times interest compounds in a period t=number of periods
Congruent Triangles
AA(Angle-Angle) If two triangles have two pairs of congruent angles, then they are similar SAS (Side-Angle-Side) if two triangles have corresponding sides that are proportional in length and the angles between those sides are congruent, then the angles are similar SSS If all the sides of one triangle are each proportional in length to the sides of another triangle, then the triangles are similar
Reflex Angles
Angles that have measures greater than 180 degrees and less than 360 degrees
Arc
Any part of the circumference. The curved portion between the points of a circle.
Triangles ΔABC and ΔPQR are shown below. The given side lengths are in centimeters. The area of ABC is 30 square centimeters. What is the area of ΔPQR, in square centimeters?
Area Triangle A = 30 so 0.5*xysin70 = 30 xysin70 = 60 xy = 63.85 Then triangle B, Area = 0.5*xysin110 = 0.5*63.85*sin110 = 30 X and Y are the same in both triangles. The reason why you get the same area is because sin110 = sin 70.
y=ax^2+bx+c STANDARD FORM
C is the y-int =b/2a is the vertex
Pythagorean Triples
Common ratios to find sides. Ex: 3:4:5 Ex: 6:8:10 (2 times the first one) Ex: 9:12:15 (3 times the first one) Ex: 27:36:45 9 times the first one) Ex: 5:12:13 Ex: 10:24:26 (2 times the first one) Ex: 15:36:39 (3 times the first one) Ex: 50:120:130 (10 times the first one)
Law of Cosines
C²=a²+b²-2abcosc
Diagonal of a Cube
Diagonal = side√3 D=s√3
Diagonal of a Square
Diagonal=side(√2)
Probability Distribution Table
Expected value: multiply data by probability, then add all up
Remainders
Find the remainder of 13 when divided by 5. 13/5=2.6 2.6-2=.6 .6 x 5=3
A perpendicular bisector
In an isosceles triangle, a segment from the non equal side to the opposite vertex is perpendicular to that side.
Facts about lines
Parallel lines: have the same slope Perpendicular lines: negative reciprocal/ opposite reciprocal slopes Also****when a problem gives you a point and an equation, plug in the point into the equation.
Multiples
Multiples of a number are always equivalent to that number or larger
Positive and Negative
Negative x Negative=Positive Negative x Positive= Negative
Even and Odd
Odd x Odd=Odd Odd x Even=Even Even x Even= Even Even+/- Odd=Odd Odd+/- Odd=Even Even +/- Even=Even
Vertical Angles
Opposite angles with be equal to each other.
Natural
The counting numbers {1, 2, 3, ...} are commonly called natural numbers
Randomization
The more randomized an experiment is, the stronger the conclusion that can be drawn
Whole
The numbers {0, 1, 2, 3, ...}.
Sector
The part of a circle that looks like a piece of pie. A sector is bounded by 2 radii and an arc of the circle.
Supplementary Angles
Two angles whose sum is 180 degrees.
Complementary Angles
Two angles whose sum is 90 degrees
Volume of a Sphere
V=4πr^3/3
Cylinder
Volume: πr²h Surface Area: 2πrh+2πr²
Rectangle
Volume:lwh surface area=2lw+2lh+2wh
In a piggy bank, there are pennies, nickel, dimes and quarters that total $2.17 in value. If there are 3 times as many pennies as there are dimes, 1 more me tan nickels, and 2 more quarters than dimes, then how many pennies are in the piggy bank.
Work backwards and look at answer choices. A) 12 B)15 C) 18 D)21 E)24 Start with C, if there are 18 pennies, then there are 6 dimes, 5 nickels, 8 quarters. the total $3.03 This is too high, so not only is C incorrect, but so are D and E. Do the same thing with A and B.
Real number
You can ignore this term, it just means any number without an "i" to mean "imaginary number." Examples: −3, −2.2, 0, 2.75, 3.1415926535897932348...(pi again!), and 2⎯⎯√
Percentage shortcut
a meal is $5.40. What will be the price with a 5% meal tax added to the total. Multiply 5.40 x 1.05.
Regular polygon
a regular polygon is one in which all sides are the same length and all angles have the same measure
Laws of Sines
a/SinA=b/SinB=c/sinC
To say that a variable falls into a particular range
abs(variable-middle range)<distance from middle to ends of range this formula tells how far from a central value you can get before being outside of the desired range Variable: w desired range: 143<w<181 middle value: 162 distance from middle to ends: 19 abs(w-162)<19 what this is saying: it is the ideal eating weight of a human, w, is less than 19 pounds away from 162
Arithmetic
add or subtract then same number between terms
When you multiply two numbers with common bases,
add the exponents.
Integers
any positive or negative whole number or 0 {...,-3,-3,-1,0,1,2,3,4...)
You can add or subtract square roots only when the numbers under the square root
are the same
Chord Problems
draw right triangle make radius the hypotenuse
at least
greater than or equal to
The Greatest Common Factor (GCF)
is the largest integer that will divide evenly into any two or more integers. --For example, the GCF of 24 and 36 is 12, because 12 is the largest integer that will divide evenly into both 24 and 36.
The Least Common Multiple (LCM)
is the smallest integer into which any two or more integers will divide evenly. --The LCM of 24 and 36 is 72, because 72 is the smallest integer into which both 24 and 36 will divide evenly.
at most
less than or equal to
Geometric
multiply or divide by the same number between terms
When you raise an exponential number to a power,
multiply the exponents
Probabilities on calc
nPr= permutations (order matters) nCr=combinations (order does not matter)
Collinear
points that lie on the same line
Sample size
rule of them: if your sample size is bigger than 100, it is probably fine
Proportions
solve by cross multiplying
Standard Deviation
standard deviation is a measure of variability, or how far values are from the mean. STANDARD DEVIATION MEANS SPREAD!!! Ex: A={1,2,3,4,5}. B={1,1,3,5,5} both sets have a mean of 3, but the values in B are slightly farther from the mean than the values in A. So set B has higher variability and a higher standard deviation.
When you divide two numbers with common bases,
subtract the exponents
ANY POLGYON
the sum of the measures of the exterior angles of any polygon is 360
Cube
volume: s^3 Surface area: 6s^2
Inscribed angle
when two chords intersect on a circle, they create an inscribed angle, the measure of an inscribed angle is half the measure of the central angle that goes to the same intercept arc.
....
when you subtracts to logs, it becomes log₂(24/3) log₂(8) and 2³=8, so 5³=125
The formula for the area of a circle is
πr²
