1 of 6 FORMULAS to UNDERSTAND & MEMORIZE Now we come to the part where you need to just bear down and memorize. To make the process a bit simpler, I am providing all of the key info that they re going to make sure you know on the ACT math section. So dig in and memorize or re-memorize all of this STUFF! Algebra Formulas The Four Steps in the Order of Operations: 1. Parentheses, mean do all operations inside enclosure marks ( ) { } [ ] before you do anything else. 2. Work out Exponents or Radicals, whichever comes first, as you read the expression from left to right. 3. Do all Multiplication or Division operations,, whichever comes first as you read the expression from left to right. 4. Finally do all Addition or Subtraction operations, whichever comes first, as you read the expression from left to right. Factoring Patterns: Factoring a Perfect Square Trinomial: a 2 + 2ab + b 2 = (a + b) 2 or a 2 2ab + b 2 = (a b) 2 Factoring a Difference of Two Squares: a 2 b 2 = (a + b)(a b) Arithmetic Sequence: For a sequence with first term t 1 and n th term t n, and with a common difference between any two consecutive terms of d, the formula for finding the t n = t 1 + (n 1)d n th term is this:
2 of 6 Exponent Rules: Radical Rules: ( ) x a x a y = a x+y x x = x a x y y = a a x a y = ax y x 2 = x y ( a ) x = a x y ( a x ) y = a x y x x = x ( ab) x = a x b ( x x ) 2 = x x a b = ax b x a b = a b a x = 1 a x a b = a b 1 a x = ax a x b y = by a x a b x = bx a x a b = a b = a b a b Coordinate Plane Formulas For any two points: P 1 ( x 1, y 1 ), and P 2 ( x 2, y 2 ) The distance, d, between the points: d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2
3 of 6 The midpoint between the points: x 1 + x 2 2, y 1 + y 2 2 The slope of a line through the points: m = y 2 y 1 x 2 x 1 Slope-Intercept Equation of a Line: where m = the slope, and b = the y-intercept Point-Slope Equation of a Line: where m = the slope of the line, and y = mx + b y y 1 = m(x x 1 ) (x 1, y 1 ) is any point on the line. Vertex Formula of a Parabola: y = a(x h) 2 + k with center (h, k) Intercept Formula of a Parabola: with x-intercepts p and q Standard Formula of a Parabola: y = a(x p)(x q) y = ax 2 + bx = c ( ) 2 + ( y k) 2 = r 2 ( ) r Circle on the Coordinate Plane: x h where h,k is the circle s center, and is the circle s radius. Data Analysis Formulas Mean = (Sum of Data Points) (# of Data Points) Sum of Data Points = (Mean) x (# of Data Points) Median = Data Point that lies in the Middle of the Set of Data Points when the Data Points are Arranged in Numerical Order Mode = Data Point(s) that Occur(s) Most Frequently Range = (Value of Greatest Data Point) (Value of Least) Data Point)
4 of 6 Plane Geometry Facts & Formulas Area of a Rectangle = base x height, or Area of a Rectangle = length x width Area of a Triangle = 1/2 (base) x (height) Area of a Parallelogram = (base) x (height) Area of a Rhombus = 1/2 (diagonal 1) x (diagonal 2) Area of a Kite = 1/2 (diagonal 1) x (diagonal 2) Area of a Trapezoid = 1/2 (base 1 + base 2) x (height) Area of a Circle = πr 2 Circumference of a Circle = 2πr Angle vocabulary: Complementary angles add up to 90 degrees. Supplementary angles add up to 180 degrees. When two lines intersect, the adjacent angles are supplementary. When two lines intersect, the angles that are across from each other (the vertical angles) are equal. Sum of Interior Angles of a Triangle = 180 degrees Sum of Interior Angles of a Convex Quadrilateral = 360 degrees Sum of Exterior Angles of a Convex Polygon = 360 degrees Sum of Measures of Angles of any Convex Polygon with n sides = (n 2)180 degrees A Regular Polygon is a polygon with all sides congruent and all angles congruent. Interior Angle of a Regular Convex Polygon with n sides = (n 2)180 / n degrees Measure of an Inscribed Angle of a Circle = 1/2 the Measure of the Arc that the Angle Cuts Off Any Angle Inscribed in a Semicircle is a Right Angle Pythagorean Theorem: In any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. A Prism is any 3-dimensional figure that has two bases identical in size and shape and sides that are perpendicular to those bases. Examples: right rectangular prisms (boxes), cylinders (cans), triangular prisms (figure with a triangle at both ends and straight faces connecting the triangles). Volume of any Prism = (area of the base) x (height)
5 of 6 Volume of Cones and Pyramids = 1/3 x (area of base) x (height) Note: You can recall the volume formula for cones and pyramids this way. Cones and pyramids come to a point, unlike prisms, which have equal-sized bases. So it s natural that the volume of cones and pyramids will be less than the volumes of their corresponding prisms. That s why the volume formulas for cones and pyramids is 1/3 the size of their corresponding prisms. To get the volumes of cones and pyramids, just get the volume of the prism (area of base x height), then multiply that by 1/3. That s the same as dividing by 3. Imaginary and Complex Numbers Imaginary Numbers: The key imaginary number is i. Here is the value of i and its first powers: i = 1 i 2 = 1 i 3 = i i 4 = +1 - - - - - - - - i 5 = i 1 = 1 i 6 = i 2 = 1 and the pattern repeats with i 7 = i 3, i 8 = i 4, and so on. Complex Numbers: The general form of a complex number is a + bi, where a and b are real numbers and i is the imaginary number. Examples of complex numbers: 2 + 3i, 5 + 7i, 2 3 + 8 3i, 0.3 1.7i, etc. To combine complex numbers, just combine the real number parts and separately combine the imaginary number parts. ( ) + ( 3+ 4i) = ( 2 + 3) + (5 + 4)i = 5 + 9i Example: 2 + 5i Trigonometry Facts & Formulas SOH-CAH-TOA gives you the basic info for finding the value of the sine, cosine or tangent functions. SOH tells you that Sine = Opposite/Hypotenuse CAH tells you that Cosine = Adjacent/Hypotenuse TOA tells you that Tangent = Opposite/Adjacent Law of Sines: a sin A = b sin B = c sinc
6 of 6 Law of Cosines (for finding a missing side, given SAS): a 2 = b 2 + c 2 2bc cos A Law of Cosines (for finding a missing angle, given SSS): cos 1 A = b2 + c 2 a 2 2bc