The Ultimate Formula Sheet for ACT Math

The Ultimate Formula Sheet for ACT Math The ACT does not provide any formulas. Be prepared by making sure to have these ones memorized. Fractions, Decimals, & Percentages: (for this section, r is the percent in decimal form) Fraction = Simple Interest: = A P(1 + rt ) part part ; Percent = whole 100 old − new Percent Increase or Decrease: old Interest Compounded Annually: = A P(1 + r )t ×100% Increase by a percent: multiply by (1 + r ) Interest Compounded n times per year: r  A P 1 +  = n   nt Decrease by a percent: multiply by (1 − r ) Rates, Ratios, & Proportions: General form of a conversion factor:  ending _ units     starting _ units   12inches  Example: 10feet   = 120inches  1foot  (Concentration of A x Volume of A) + (Concentration of B x Volume of B) = Final concentration (Vol. of A + Vol. of B) Distance = Rate x Time Exponents, Roots, & Polynomials: Multiplication Rule for Exponents: ab ⋅ ac = a b +c Division Rule for Exponents: ab = a b −c ac ( ) Power Rule for Exponents: ab c = abc b Fractional Exponents: a c = c ab or i= ( a) c b −1 ; i 2 = −1 ; i 3 = −i ; i 4 = 1 i 4 n = 1 ; i 4 n+1 = i ; i 4 n+2 = −1 ; i 4 n+3 = −i Complex Conjugates: (a + bi )(a − bi ) 1 Negative Exponents: a = b a −b Parabolas: Standard Form: f ( x ) = ax 2 + bx + c ; Discriminant = b2 − 4ac ; Pos=2 real roots  b  b  vertex=  − , f  −   ;  2a  2a   Zero= 1 real root; Neg=2 imaginary roots y-intercept = c; x-intercepts are m and n; x-intercepts = Factored Form: f ( x ) =a( x − m)( x − n) ; −b ± b2 − 4ac 2a Sum of solutions = x-coordinate of vertex = −b a m+n 2 Vertex Form: f ( x ) = a( x − h)2 + k ; vertex = (h, k ) . Difference of Squares: a2 − b2 =(a + b)(a − b) Sum of Cubes: a3 + b3 =(a + b)(a2 − ab + b2 ) Difference of Cubes: a3 − b3 =(a − b)(a2 + ab + b2 ) Perfect Square Trinomial: a2 + 2ab + b2 =( a + b ) and a2 − 2ab + b2 =( a − b ) 2 2 b b  Completing the Square: x 2 + bx +   =  x +  2 2  2 2 Graphing Lines: Slope Formula: m = y 2 − y1 x2 − x1 Standard Form: Ax + By = C Slope-Intercept Form:= y mx + b Slope of horizontal line = 0 Point-Slope Form: y − y1 = m( x − x1 ) Slope of vertical line = undefined Distance Formula: d= ( x2 − x1 )2 + (y 2 − y1 )2  x + x y +y  Midpoint Formula: M =  1 2 , 1 2  2   2 Parallel lines: equal slopes ⊥ Lines: slopes are opposite reciprocals Parent Graphs & Transformations: y=x y= x y = x2 y = x3 y = ax y= x Transformation f (x) + k f (x) − k f ( x + h) f ( x − h) −f ( x ) cf ( x ) Visual effect Shift up by k units Shift down by k units Shift left by h units Shift right by h units Reflect over the x axis (flip upside down) Stretch vertically by a factor of c (becomes skinnier) 1 f (x) c Data & Probability: Average = Shrink vertically by a factor of c (becomes fatter) sum _ of _ items number _ of _ items Median = center data point Mode = most frequent data point Probability that independent events A and B will both happen: P( A = B) P( A)× P(B) Probability that either A or B will happen: P( A  B) = P( A) + P(B) − P( A  B) n Range = maximum – minimum Probability = Expected Value: E( x ) = ∑ xi P( xi ) i =1 desired _ outcomes possible _ outcomes Angles: Vertical ∠’ s are ≅ ∠’s that form a linear pair are supplementary (add up to 180°) Triangles: 1 Area of a Triangle: A = bh 2 ∠’ s that form a circle add up to 360° When ∥ lines are cut by a transversal, all acute ∠’ s are ≅ and all obtuse ∠’ s are ≅ Special Right Triangles: The three ∠’ s of a ∆ add up to 180° An exterior ∠ is equal to the sum of the two remote interior ∠’ s Pythagorean Theorem: a2 + b2 = c2 Pythagorean Triples: 3-4-5 and 5-12-13 Circles: Area of a Circle: A = π r 2 A radius and tangent make a right ∠ Circumference of a Circle: C = 2π r A central ∠ is double the inscribed ∠ x arc = 360 circumference x sec tor = 360 area _ of _ circle and where x = central angle Polygons: (for this section, n is the number of sides) Area of a Rectangle: A = lw Area of a trapezoid: One int. ∠ of a regular polygon: 1 (b1 + b2 )h 2 # of diagonals: Sum of the exterior angles: 360° 180(n − 2) n n(n − 3) (convex only) 2 Sum of the interior angles: 180(n − 2) Properties of Parallelograms: 1. Opp sides are ∥ and ≅ 5. Diagonals bisect each other 3. Consec ∠’s are supplementary If they are ⊥ it is a rhombus If they are ≅ it is a rectangle 2. Opp ∠’ s are ≅ 6. Area = base × height 4. Each diagonal forms a pair of ≅∆’s Solids: Volume of a Rectangular Prism (Box): V = lwh 4 Volume of a Sphere: V = π r 3 3 Surface Area of a Box: SA= 2(lw + lh + wh) 1 Volume of a Cone: V = π r 2 h 3 Volume of a Cylinder: V = π r 2 h Surface Area of a Cylinder:= SA 2π r 2 + 2π rh 1 Volume of a Pyramid: V = lwh 3 Trigonometry: sin = opp hyp cos = 360°=2π radians adj hyp tan x = tan = sin x cos x opp adj csc( x ) = sec( x ) = sin2 x + cos2 x = 1 a b c Law of Sines: = = sin A sin B sin C y = sin( x ) 1 sin( x ) 1 cos( x ) cot( x ) = 1 tan( x ) = sin( x ) cos(90 − x ) Law of Cosines: a2 = b2 + c 2 − 2bc cos( A) y = cos( x ) y = tan( x ) If y A sin(Bx − C ) + D (also for cos, csc, and sec) = Amplitude: A Period: 2π B Phase Shift: C B Vertical Shift: D Phase Shift: C B Vertical Shift: D If y A tan(Bx − C ) + D (also for cot) = Amplitude: none Period: π B Sequences and Series: where a1 = first term, n = number of terms, d = common difference, r = common ratio Arithmetic sequence: an = a1 + (n − 1)d Sum of an arithmetic series:= Sn n ( a1 + an ) 2 Geometric sequence: an = a1 r n−1 Sum of a geometric series: Sn = a1 (r n − 1) r −1 Logarithms: If log b a = x , then b x = a   Vector Addition: a + b= log b a = log a log b a2 + b2 + 2ab cosθ Matrix Multiplication: Only possible when columns of first = rows of second  A B   E F   AE + BG AF + BH   × =    C D   G H   CE + DG CF + DH  A B Determinant of  =  AD − BC C D Conic Sections: Circle: ( x − h)2 + (y − k )2 = r 2 , where (h,k) is the center and r is the radius Ellipse: ( x − h)2 (y − k )2 + = 1 where (h,k) is the center, 2a is the horizontal axis, and 2b is the vertical axis a2 b2 2 2 Horizontal Ellipse: a= b2 + c 2 Vertical Ellipse: b= a2 + c 2 where c is the distance from center to focus Horizontal Hyperbola: ( x − h)2 (y − k )2 (y − k )2 ( x − h)2 Vertical Hyperbola: − = 1 − = 1 a2 b2 a2 b2