math

circumference of a circle

C = 2 π r, where C is the circumference and r is the radius.

rotations

In a rotation of 90° about the origin, point (A,B) --> (-B,A). In a rotation of 180°, point (A,B) --> (-A,-B). In a rotation of 270°, point (A,B) --> (B,-A). Hint! If you have trouble visualizing what a rotated figure would look like, simply rotate your test book the number of degrees mentioned in the question.

converting degrees to radians

To convert degrees to radians, multiply by π/180. Example: Convert 270° to radians. 270 ×π/180 = 3π/2

area of a circle

A = πr^2, where A is the area and r is the radius.

remainders

A remainder is the amount "left over" after dividing one integer by another to produce an integer quotient. Example: 43/3=14 R3

prime numbers

An integer greater than 1 that has no positive divisors other than 1 and itself. Important to know! 1 is NOT a prime number. 2 IS a prime number. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Even and Odd Integers

Even integers can be divided evenly by two; odd integers cannot. Examples: 2, 4, 6, 0, -2, -4, 6 are all even. 1, 3, 5, -1, -3, -5 are all odd.

FOIL Method.

FOIL helps us remember to how to distribute two binomials. FOIL stands for First (multiply the first terms in each), Outer (multiply the outer terms in each), Inner (multiply the inner terms in each), Last (multiply the last terms in each). Example: (2x + 4)(3x + 3) = (2x)(3x)+(2x)(3) +(4)(3x) + (4)(3) = 6x2 + 6x + 12x + 12 = 6x^2 + 18x + 12.

slope-intercept form

One way of writing the equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Important to know! This is the most useful form of a straight-line equation on the ACT, as it immediately gives information about slope and y-intercept. If a line equation is not in this form, try to shift it around to this form.

Order of operations (PEMDAS)

PEMDAS is an acronym that helps you remember the order you must do operations in on an arithmetic problem: 1. Parentheses (any grouping symbols, including brackets and fraction bars) 2. Exponents 3. Multiplication 4. Division 5. Addition 6. Subtraction. Good to know! You may remember this better as "Please Excuse My Dear Aunt Sally."

reciprocal identities

Secant θ (sec) = 1/cos θ. Cosecant θ (csc) = 1/sin θ. Cotangent θ (cot) = 1/tan θ.

similar triangles

Similar triangles are triangles that have corresponding equal angles and proportional sides. Think of them as mama and baby triangles that look alike, only one is bigger. Important to know! You can find the missing sides of similar triangles by setting up proportions. EX: Triangles 1: one leg is 6 and the other is 3 2nd triangle: one leg is 3 and the other is x 6/4=3/x 6x=12 x=12

equation of a circle

The equation of circle is (x-h)62 + (y-k)62 = r62, where (h, k) is center of the circle and r is the radius.

simple interest

The formula for simple interest, where A = the final value, P = the principal (original) amount, r = interest rate per period, and t = number of time periods, is A = P(1 + rt). Example: If Susan borrows $4500 at a 10% annual interest rate, in 5 years, how much will she owe if she hasn't made any payments? Answer: A = 4500(1 + 0.10 × 5) = $6750.

dividing power with different bases

To divide exponents with different bases, divide the bases and keep the exponent the same. Examples: y^4 ÷ z^4 = (y/z)^4 ; 6^4 ÷ 3^4 = (6/3)^4 = 2^4 = 16

multiplying and dividing fractions

To multiply, multiply the top numbers by each other and the bottom numbers by each other. To divide, 'flip and multiply": flip the second fraction (turn it into its reciprocal) and multiply it by the first fraction. Simplify if necessary. Examples: 3/4 x 5/6= 5/8 3/4 / 5/6=3/4 x 6/5= 18/20=9/10

reflections

When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite. When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite. Good to know! When you reflect a point, line, or curve across a line, think of the line as serving as a mirror. Where would the reflection of each point be in the mirror?

converting decimals to fractions

Write down the decimal divided by 1, then multiply the top and bottom by 10 for every number after the decimal point. For example, to convert 0.62 to a fraction, multiply 0.62/1 by 100/100, which equals 62/100. Now reduce: 62/100 = 31/50. Tip: It will help to memorize the decimal equivalents of common fractions. Hint: The simplest method is to use your calculator. If you have something like a TI-83, press MATH then >FRAC.

cylinder

a cylinder is a three dimensional shape with two identical bases that are circular or elliptical. Good to know! The volume of a right cylinder (one that stands straight up) is πr^2 × height.

logarithms

are the inverses of exponents; they undo exponentials. y = bx is equivalent to logb(y) = x. Example: Evaluate log39. log39 --> 9 = 3^x, x = 2.

Standard form of the equation of a line

ax + by = c Important to know! This is the form that you will most often see on ACT questions, but, for the purposes of the test, it is not as useful as slope-intercept form.

quadratic formula

can be used to find solutions to quadratic equations (ax2 + bx + c = 0). The quadratic formula looks like what is below. Plug in the values and solve for x! (The plus-minus sign means you'll get two results). x=-6 + or - square root of b^2-4ac/2a

right triangle

has one angle of 90°.

adding matrices

in order to add matrices, the matrices need to have the exact same dimensions. If this is the case, then add or subtract the corresponding pairs of entries.

inequalities

inequalities are solved when the variable is isolated on one side of the inequality. Example: 2x+3>4x-2 2x+5>4x 5>2x 5/2>x

x- and y-intercepts

intercepts of a graph are where the line or curve crosses the x- or y-axis. Important to know! To find the y-intercept, plug in 0 for x and solve for y. To find the x-intercept, plug in 0 for y and solve for x.

scientific notation

is a method to handle very large or very small numbers without writing out all the place holding zeros. Simply count how many spaces you are moving the decimal point to the right or the left. There should always be one digit before the decimal point. Examples: 5,340,000,000 is 5.34 × 10^9 and 0.0000000000425 is 4.25 ×10^11

rounding

is a method used to make numbers shorter and simpler by leaving off some of digits of smaller values. For the digit you are rounding, if it is less than 5, round down and equal to or greater than 5, round up. Examples: 23,459 rounded to the thousands place is 23,000 and rounded to the hundreds place is 23,500. 0.089 rounded to the tenths place is 0.1.

SOHCAHTOA

is a mnemonic that helps us remember how to compute the sine, cosine and tangent of an angle in a right triangle. Sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent.

regular polygon

is a multi-sided shape that has all equal angles and sides of the same length.

perfect squares

is a number that can be expressed as the product of two equal integers. Examples: 1: (1 × 1), 4: (2 × 2), 9: (3 × 3), 81: (9 x 9), 625: (25 × 25)

trapezoid

is a quadrilateral with exactly one pair of parallel sides. Good to know! The area of a trapezoid = (length of base A + length of base B)/2 × height.

area of a parallelogram

is a shape with 2 sets of parallel sides. The area of a parallelogram is base × height. Important to know: Remember the height may not be the length of a side!

parabola

is a special curve shaped like an arc. The standard form of a parabola is y = ax2 + bx + c. The vertex form is y = a(x-h)2 + k, where (h,k) is the vertex. If a > 0, then the parabola opens upwards. If a < 0, then it opens downward. Good to know! If the absolute value of a < 1, the graph of the parabola widens. If the absolute value of a > 1, the graph becomes narrower.

ratios

is an expression of how one number compares to the size of another number. Example: There are 10 boys and 8 girls on the soccer team, so the ratio of boys to girls on the is 5:4.

absolute value

is the distance on the number line from a given number to zero. To find the absolute value of a number, simply remove the negative sign in front of a number. Absolute value notation looks like this: |-2|. Example: the absolute value of -15, or |-15|, is 15.

exponents

of a number tells you how many factors of the number are being multiplied together. Example: 4^3 = 4 × 4 × 4 = 64

square root

of a number x is a number y whose square (y × y) equals x. Example: 3 is a square root of 9 since 3 × 3 = 9.

law of sines

sin A/a=sinB/b=sin C/c

dividing powers with same base

to divide exponents with the same base, subtract the exponents. Example: y^5 ÷ y^2 = y3.

absolute value inequalities

to solve absolute value inequalities, you need to find two solutions because the expression inside the absolute value could be either positive or negative. Example: /4x+2/>6 -6<4x+2<6 you can break it into two inequalities now -6<4x+2 -8<4x -2<x 4x+2<6 4x<4 x<1 -2<x>1

least common multiple

The smallest positive integer that is divisible by two numbers. To find the LCM: Method 1: list the prime factors of each number and multiply each factor by the greatest number of times it occurs in either number. Method 2: list the multiples of a number and find the first one that is the same. Example for Method 1: The LCM of 8 (2 × 2 × 2) and 14 (2 × 7) is 56 (2 × 2 × 2 × 7). Example for Method 2: 8: 8, 16, 24, 32, 40, 48, 56. 14: 14, 28, 42, 56. So the LCM is 56.

adding and subtracting exponents

To add or subtract the same base, the powers must be the same. The exponent does not change, but the bases are added together. Examples: x^3 + x^3 = 2x3 but x^2 + x^3 does not equal x^5.

percent increase and decrease

To calculate percent increase or decrease, find the difference between the two numbers you are comparing, then divide the result by the original number and multiply by 100. Percent Increase = (New Number - Original Number)/Original Number x 100. Percent Decrease = (Original Number - New Number)/Original Number × 100. Examples: percent increase from 24 to 36: 36-24/24x100=50% Percent decrease from 36 to 24: 36-24/36x100 similar to 33.3%

converting radians to degrees

To convert radians to degrees, multiply by 180/π. Example: Convert 3π/2 to degrees. 3π/2 x 180/π = 270°.

graphing inequalities

To find a solution to a linear inequality on the coordinate plane, first find the equation as if it were a line. If the inequality says y < that line, then shade below that line for the solution. If the inequality says y > that line, then shade above that line for the solution. Good to know! If the inequality is "less than or equal to" or "greater than or equal to" then the line (and points on it) is included in the solution.

multiplying powers with different bases and same exponents

To multiply exponents with different bases and the same exponents, multiply the bases together and keep the exponent the same. Examples: y^4 × z^4 = (yz)^4 ; 2^2 × 3^2 = (2 × 3)^2 = 6^2 = 36.

multiplying powers with same base

To multiply powers with the same base, add the exponents. Example: y^3 × y^2 = y^5.

rationalizing radical expressions

Rationalizing is the process of getting rid of all the radicals that are in the denominator (bottom number) of a fraction. To rationalize, simplify the radicals and fractions if necessary, and then multiply both the top and bottom by a radical that will get rid of the radical in the denominator. Example: 2/ square root of 6=2/square root of 6 x squre root of 6/square root of 6=2xsqaure root of 6/6

inequalities with negative numbers

Remember when multiplying or dividing by negative numbers in an inequality that you need to reverse the direction of the inequality! Example: -2y>3 y<-3/2

converting fractions to decimals

Find a number you can multiply the denominator by to make it 10, 100, 1000, and so on. For example, to convert ¼ to a decimal, multiple the top and bottom by 25. This equals 25/100. Take the number in the numerator and put the decimal point 2 spaces from the right, so ¼ = 0.25. Tip: It will help to memorize the decimal equivalents of common fractions. Hint: The simplest method is to use your calculator. If you have something like a TI-83, press MATH then >DEC.

igainary numbers pattern

For the purposes of the ACT, you should remember the pattern for imaginary numbers: i = i, i^2 = -1, i^3 = -i, i^4 = 1, then the pattern continues to repeat. Example: What is i^23? (Find the closest multiple of 4 that is less than 23. It is 20. i20=i^4. Then count up to 23: i^21=i1^, i2^2=i^2, and finally, i^20=i^3, which equals -i, so i^23 = -i.

Operations of Even and Odd Integers

adding, subtracting, multiplying, and dividing even and odd numbers always yields the same results regarding whether the result is odd or even. For example, even + even = even and even + odd = odd. Important to Know! If this comes up on the ACT, simply test an easy case. Would an odd - odd be even or odd? Try 5 - 3 , which equals 2, so odd - odd is always even.

inverse trig functions

are used to obtain an angle measure from any of the angle's trigonometric ratios. Written, for example, as sin-1(½) = θ or arctan 2⁄3 = θ. Example: What is arcsin(½)? (The ACT will give you a table of values or a diagram to solve this, or you can use your calculator) Answer: 30°.

similiar triangles

Similar triangles are triangles that have corresponding equal angles and proportional sides. Think of them as mama and baby triangles that look alike, only one is bigger. Important to know! You can find the missing sides of similar triangles by setting up proportions.

unit circle

is a circle with a radius of 1 that is useful in trigonometry. Angles are measured starting from the positive x-axis in quadrant 1 and continuing counterclockwise. For the ACT, you should definitely know the radian measures of the axes and in which quadrants the trig functions are positive or negative. Important to know! The mnemonic All Students Take Calculus can help you remember where functions are positive: Q1: all, Q2: sine, Q3: tangent, Q4: cosine. Quadrant one if top right, quadrant 2 is top left, quadrant 3 is bottom left and quadrant 4 is bottom right

sector

is a fraction of the area of a circle. It is a "slice of the pie." To find the area of a sector, you can set up a proportion of sector area/entire area to central angle/360. Example: What is the length of arc ST if the radius of the circle is 3? Remember area is π r2 , so the area of the circle is 9/8π. Ex radius is 3 and angle is 45 degrees in the pie x/9pi=45/360 x/9pi=1/8 8x=9pi x=9/8pi

arc

is a fraction of the circumference of a circle. To find the length of an arc, you can set up a proportion of arc length/circumference to central angle/360. Example: What is the length of arc ST if the radius of the circle is 3? Remember circumference is 2πr, so the circumference of the circle is 6π. EX: radius is 3 and angle is 45 degrees x/6pi=45/360 x/6pi=1/8 8x=6pi x=3/4pi

matrices

is a rectangular array of numbers defined by the number of rows and columns it contains, in that order (a 3 x 2 matrix has 3 rows and 2 columns). Example of a 2 x 3 matrix: (3 4 7) (2 3 1)

functions

is a special relationship where each input has a single output. Think of functions as machines that spit out certain values for each input. Functions are often written as f(x) where x is the input value. Hint: it might be easier for you to think of f(x) as a y when dealing with a basic equation Example: If f(x) = x + 2, what does f(4) equal? When x = 4, f(4) = 4 + 2, so f(4) = 6, making 6 our "output."

sphere

is a three-dimensional circle. Imagine the surface of a ball. Good to know: the volume of a sphere = 4/3πr^3

permutation

is all possible arrangements of a collection of things (order matters). nPr = n!/(n-r)! Example: The arrangement of 6 students in 3 different rows can be done in 6P3 ways. 6P3 = 6!/(6-3)!, which is 120.

obtuse triangle

is an angle measuring greater than 90° but less than 180°.

acute angle

is an angle measuring less than 90°.

combination

is an arrangement of things in which order does not matter. nCr = n!/[r!(n-r)!] The possible combinations of 8 students in 4 different teams is 8C4 = 8!/[4!(8-4)!], which is 70.

change of bas formula

is log^bx = log x / log b.

scalar multiplication of a matrix

is multiplying a matrix by an ordinary number. To do this, multiply each entry by that number. Ex: 2x(3 4 7)=(6 8 14) (2 3 1)=(4 6 2)

imaginary numbers

is one that when squared gives a negative result. The imaginary unit is written as the letter i. You can add, subtract, multiply and divide complex numbers. Important to know! i^2 = -1. Example: 4i2 - 5i - 2i = 4(-1) - 5i - 2i = -4 - 7i

volume

is the amount of space inside a 3D object. Volume formulas you should know: Rectangular solid: V = length × width × height. Sphere: V = 4/3 πr3. Right cylinder: V = πr2h.

median

is the middle number in a set of numbers if they were put into order. If the set has an even number of items, the median is the average of the two middle numbers. The median of -23, 29, 3, 84, and -2 is 3.

mode

is the most frequently occurring number in a set of numbers. Sets can have no modes or several modes. The mode of 3, 3, 2, 7, 6, 4, 3, 2 is 3.

finding factors

is the process of listing the factors of a number, basically splitting an expression into simpler expression. Example: 2, 3, 5, 6, 10, and 15 are factors of 30 because 2 × 15 = 30, 3 × 10 = 30, and 6 × 5 = 30.

percents

means parts per hundred. When expressed as a fraction, the denominator can be expressed as 100. Examples: 75% = 75/100 23% = 23/100

weighted averages

A weighted average is an average resulting from multiplying each component by a factor indicating its importance. Example: Many of your teachers probably use weighted averages to calculate your grades: homework might be worth 20%, participation 20% and tests 60%, for example. If Angela has a homework grade of 98%, participation grade of 90%, and test grade of 82%, her final grade would be (0.20)(98) + (0.20)(90) + (0.60)(82) = 86.8.

estimating ACT figures

It's important to know that unless the figure says otherwise, ACT geometry figures are all drawn roughly to scale. You can use this knowledge to estimate side lengths and angles.

45-45-90 triangle

Memorizing the side ratios of the special right triangles can save you a lot of time on the ACT. The respective sides of 45°-45°-90° triangles are in the ratio: 1: 1: √2, or x:x:x√2.

compound interest

The formula for compound interest, A = the final value, P = the principal (original) amount, r = interest rate per period, n = the number of times per year that interest is compounded, and t = number of years, is A = P (1 + r/n)nt . Example: In 3 years, how much money will $1000 be worth if put in a savings account compounded monthly at 2% interest? Answer: A = 1000 (1 + 0.02/12)(12)(3)≅ $1062.

greatest common factor

The greatest factor that divides two numbers. To find the GCF, list the prime factors of each number, circle the pairs of factors both numbers have in common, and multiply those factors together. Example: The GCF of 12 (2 × 2 × 3) and 30 (2 × 3 × 5) is 6 (2 x 3). To check, make sure both 12 and 30 divide by 6.

mean

The mean is the average of a set of numbers. Mean = sum / the number of things. On the ACT, however, it's often more helpful to use this form: sum = mean × things. Questions will often give you the average and ask you to find the sum. Example: What is the mean of Tony's exam scores if he scored a 88, 93, 80, and 99 on his exams? Answer: Mean = (88 + 93 + 80 + 99) / 4 = 90.

prime factors

The prime factors of a number are the prime numbers that are divisors of that number. Examples: The prime factors of 15 are 3 and 5. The prime factors of 18 are 3 and 2 because 3 ×3 × 2 = 18.

reciprocal

The reciprocal of a number is 1 divided by that number. Essentially, its inverse. Example: The reciprocal of 4 is ¼ and the reciprocal of ¼ is 4.

multiples

The result of multiplying a certain number by an integer. Multiples can be positive OR negative. When you learned your times tables, you were learning multiples. 4, 6, 8, and 10 are multiples of 2 because 2 × 2 = 4, 2 × 3 = 6, 2 × 4 = 8 and so on.

slope

The slope of a line tells us the direction and the steepness of a line. You are probably familiar with it as "rise over run" (or change in y over change in x). Formula, where (x1 , y1) and (x2 , y2) are two points on the line: slope=y2-y1/x2-x1

area of a triangle

The formula for the area of a triangle is (base × height) / 2. Hint: Remember this is because a triangle can be considered to be half of a rectangle and the area of a rectangle is base × height. Example: Area of the purple triangle below = (5 × 3)/2 = 7.5.

Adding and Subtracting Fractions

To add or subtract fractions, the denominators must be the same. Find a common denominator, convert the fractions, then add or subtract the numerators. Example: 3/4 + 5/6=9/12 +10/12=19/12

dividing complex numbers

To divide a complex number, multiply the expression by the complex conjugate of its denominator. Ex: simplify 5+2i/6+3i 5+2i/6+3i x 6-3i/6-3i (30-15i+12i-6i^2)/(36-18i+18i-9i^2) remember that i^2=-1 (30-15i+12i+6)/(36-18i+18i-9)

perimeter

To find the perimeter of a figure, add up all of the side lengths. Important to know! Remember that opposite sides in a rectangle are equal to one another.

Python

a^2 + b^2 = c^2. The first leg of a right triangle squared + the second leg of the triangle squared = the hypotenuse squared. Good to know! Memorizing the common ratios of sides in a Pythagorean triple triangle can save you time on the ACT. Here are the ones to know: (3,4,5), (5,12,13), (7, 24, 25), (8,15,17).

right angle

is an angle measuring 90°.

pythagoran identity

is sin^2 θ + cos^2 θ = 1.

cube root

of a number x is a number y whose cube (y × y × y) equals x. Although we can't take a square root of a negative number, we can take a cube root of a negative number. 3 is a cube root of 27 because 3 × 3 × 3 = 27.

translation

slides every point on a figure the same distance in a certain direction. Translations in quadratics will appear on the ACT. The basic quadratic equation is f(x) = x^2 . Adding or subtracting values outside the entire expression shifts the graph up for addition and down for subtraction. Adding or subtracting values with each x value within the expression shifts the graph to the left for addition and right for subtraction. Examples: f(x) = x^2 + 2 shifts the graph up 2 units. f(x) = (x - 3)^2 shifts the graph to the right 3 units.

negative exponents

tells us how many times to divide that number. The answer will be the reciprocal of the number to the positive exponent. Example: 4^-3 = 1 ÷ 4 ÷ 4 ÷ 4 = 1/64

fundamental counting principle

the FCP is a way to figure out the total number of ways different events can occur. If there are a ways for one activity to occur and b ways for a second activity to occur, then there are a x b ways for both to occur. Example: Mattie has three pants, four jackets, and six shirts to choose from for an outfit. How many possible outfits can she make? 3 x 4 x 6 = 72 possible outfits. (All items must be distinct!)