Math ACT

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

Imaginary numbers pattern

i = i, i2 = -1, i3 = -i, i4 = 1, then the pattern continues to repeat. Example: What is i23? (Find the closest multiple of 4 that is less than 23. It is 20. i20=i4. Then count up to 23: i21=i1, i22=i2, and finally, i20=i3, which equals -i, so i23 = -i

inequality

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

Inverse Trig Functions

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)? Answer: 30°.

Logarithm

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

First 10 prime #s

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself

Converting Radians to Degrees

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

Pythagorean Identities

sin2θ + cos2θ = 1.

Pythagorean Identity

sin²θ + cos²θ = 1

Circumference of a circle

C = 2πr, where C is the circumference and r is the radius

Area or circle

A = πr2, where A is the area and r is the radius

combination

A 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.

Functions

A function 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

negative exponent

A negative exponent 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

Parabola

A 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.

Area of a parallelogram

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!

regular polygon

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

Sector

A 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.

Sphere

A sphere is a three-dimensional circle. Imagine the surface of a ball. Good to know: the volume of a sphere = 4/3πr3

Translations

A 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) = x2 . 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) = x2 + 2 shifts the graph up 2 units. f(x) = (x - 3)2 shifts the graph to the right 3 units.

Trapezoid

A 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

weighted average

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

acute angle

An acute angle is an angle measuring less than 90°.

Arc

An 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.

Prime numbers

An integer greater than 1 that has no positive divisors other than 1 and itself. 1 is NOT a prime number. 2 IS a prime number.

obtuse angle

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

Unit Circle

Aunit 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.

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 = 6x2 + 18x + 12.

Finding factors

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.

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.

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.

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.

rationalizing radicals 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. Ex) 2/√6 = 2/√6 × √6/√6 = 2√6/6

Inequalities with negative #

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

SOHCAHTOA

SIN (Opposite/Hypotenuse) COS (Adjacent/Hypotenuse) TAN (Opposite/Adjacent)

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. Example:

change of base formula

The change of base formula is logbx = log x / log b.

cube root

The 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.

Equation of a circle

The equation of circle is (x-h)2 + (y-k)2 = r2, where (h, k) is center of the circle and r is the radius.

Exponents

The exponent of a number tells you how many factors of the number are being multiplied together. Example: 43 = 4 × 4 × 4 = 64

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.

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

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 = 15

x and y intercepts

The 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.

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.

Median

The 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

The 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.

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.

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:

Square root

The 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.

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: x3 + x3 = 2x3 but x2 + x3 does not equal x5

percent increase/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 = 36-24/24 x 100= 50% Percent decrease from 36 to 24 = 36-24/36 x 100= 33.3%

converting degrees to radians

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

Dividing complex numbers

To divide a complex number, multiply the expression by the complex conjugate of its denominator.

Dividing powers with different base

To divide exponents with different bases, divide the bases and keep the exponent the same. Examples: y4 ÷ z4 = (y/z)4 ; 64 ÷ 34 = (6/3)4 = 24 = 16

Dividing Fractions

To divide, 'flip and multiply": flip the second fraction (turn it into its reciprocal) and multiply it by the first fraction. Simplify if necessary. Ex) https://magoosh-production.s3.amazonaws.com/flashcard_entries/3583/images/large_desktops.png?1454657787

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: y4 × z4 = (yz)4 ; 22 × 32 = (2 × 3)2 = 62 = 36.

Multiplying powers with the same base

To multiply powers with the same base, add the exponents. Example: y3 × y2 = y5.

Quadratic Formula

Used for equation such as ax^2+bc+c=0

Volume

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.

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 a decimal to a fraction

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 πr2 × height

permutation

a 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.

Pythagorean triple

a set of three positive integers that work in the pythagorean theorem (3,4,5) (5, 12, 13) (7, 24, 25) (8, 15,17)

Pythagorean Theorem

a2 + b2 = c2. 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).

imaginary number

an imaginary number 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! i2 = -1. Example: 4i2 - 5i - 2i = 4(-1) - 5i - 2i = -4 - 7i

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!)

Dividing powers with the same base

to divide exponents with the same base, subtract the exponents. Example: y5 ÷ y2 = y3.

Absolute value inequality

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