CBC MATHEMATICS DIVISION MATH 2412-PreCalculus Exam Formula Sheets

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1 System of Equations and Matrices 3 Matrix Row Operations: MATH 41-PreCalculus Switch any two rows. Multiply any row by a nonzero constant. Add any constant-multiple row to another Even and Odd functions Even function: f( x) = f(x) Odd function: f( x) = f(x) Graph Symmetry x-axis symmetry: if (x, y) is on the graph, then (x, y) is also on the graph y-axis symmetry: if (x, y) is on the graph, then ( x, y) is also on the graph origin symmetry: if (x, y)f is on the graph, then ( x, y) is also on the graph Function Transformations Stretch and Compress y = af(x), a > 0 vertical: stretch f(x) if a > 1 Reflections y = f(x) reflect f(x) about x-axis y = f( x) reflect f(x) about y-axis Stretch and Compress y = af(x), a > 0 vertical: stretch f(x) if a > 1 : compress f(x) if 0 < a < 1 y = f(ax), a > 0 horizontal: stretch f(x) if 0 < a < 1 : compress f(x) if a > 1 Shifts y = f(x) + k, k > 0 vertical: shift f(x) up y = f(x) k, k > 0 : shift f(x) down y = f(x + h) h > 0 horizontal: shift f(x) left y = f(x h), h > 0 : shift f(x) right Auth:C.Villarreal-Prof. CBC 015Spring

2 Formulas/Equations MATH 41-PreCalculus Slope Intercept: y = mx + b Point-Slope: y 1 y = m(x x 1 ) Slope: m = y y 1 x x 1 ; x x 1 0 Average Rate of Change: Δy = f(b) f(a), where a b Δx b a Circle: Circumference = πr = πd, Area = πr Triangle: Area = 1 bh Rectangle: Perimeter = l + w, Area = lw Rectangular Solid: Volume = lwh, Surface Area = lw + lh + wh Sphere: Volume = 4 3 πr3, Surface Area = 4πr Right Circular Cylinder: Volume = πr h, Surface Area = πr + πrh General Form of Quadratic Function: f(x) = ax + bx + c, (a 0) Quadratic Formula: x = b± b 4ac a Vertex (h, k): h = b k = a(h) + b(h) + c, a or ( b b b, f ( )), or (, 4ac b ) a a a 4a Axis of symmetry: x = h Vertex Form of Quadratic Function: f(x) = a(x h) + k Polynomial function: f(x) = a n x n + a n 1 x n a 1 x 1 + a 0 Polynomial graph has at most n 1 turning points. Remainder Theorem If polynomial f(x) (x c), remainder is f(c). Factor Theorem If f(c) = 0, then x c is a linear factor of f(x). If x c is a linear factor of f(x), then f(c) = 0. Rational Zeros Theorem vertex (h, k) Possible rational zeros: ± p q, where p is a factor of a 0 and q is a factor of a n. Auth:C.Villarreal-Prof. CBC 015Spring

3 MATH 41-PreCalculus Intermediate Value Theorem(for continuous function f(x)) If a < b and if f(a) and f(b) have opposite signs, then f(x) has at least one real zero between x = a and x = b. Conjugate Pairs Theorem For polynomial functions f(x) with real coefficients: If x = a + bi is a zero of f(x), then x = a bi is also. Rational function: f(x) = p(x), p(x) and q(x) polynomials, but q(x) 0. q(x) Vertical Asymptote: x = zero of denominator in reduced f(x) Horizontal Asymptote: y = 0 if degree of p(x) < degree of q(x) y = Oblique Asymptote: leading coefficient of p(x) leading coefficient of q(x) y = quotient of p(x) q(x) Composite Function (f g)(x) = f( g(x) ) Exponential Function: f(x) = a x If a u = a v, then u = v Logarithmic Function: f(x) = log a (x) if degree of p(x) = degree of q(x) if degree of p(x) > degree of q(x) log a (1) = 0, log a (a) = 1, a log a(m) = M, log a (a p ) = p log a ( M N ) = log a (M) + log a (N) log a ( M N ) = log a(m) log a (N) log a (M p ) = p log a (M) If log a (M) = log a (N), then M = N. If M = N, then log a (M) = log a (N). Change of Base formula log a (M) = log(m) log(a) or log a (M) = ln(m) ln(a) Auth:C.Villarreal-Prof. CBC 015Spring

4 Exponential Models Formulas Simple Interest: I = Prt MATH 41-PreCalculus Compound Interest: A = P (1 + r n )n t Continuous Compounding: A = Pe r t Effective Rate of Interest: Compounding n times per year r eff = (1 + r n )n 1 Compounding continuously per year r eff = e r 1 Growth & Decay: A(t) = A 0 e k t Newton s Law of Cooling: u(t) = T + (u 0 T)e k t Logistic Model: P(t) = Sequences and Series c 1+ae b t n! = n(n 1)(n ) (3)()(1) P(n, r) = n! C(n, r) = (n r)! Arithmetic Sequence: n th term n! r!(n r)! a n = a 1 + (n 1)d n Sum of first n terms S n = k=1 (a 1 + (k 1)d) = n (a 1 + a n ) or S n = n k=1 (a 1 + (k 1)d) = n (a 1 + (n 1)d). Geometric Sequence: n th term a n = a 1 (r) n 1 Sum of first n terms S n = n k=1 a 1 r k 1 = a 1 1 rn 1 r for r 0,1 Geometric Series: k=1 a 1 r k 1 = a 1 1 r Binomial Theorem: (x + a) n = n j=0 ( n j ) xn j a j if r < 1 = ( n 0 )xn + ( n 1 )xn 1 a + + ( n n 1 )xan 1 + ( n n )an Auth:C.Villarreal-Prof. CBC 015Spring

5 Trigonometry Circular Measure and Motion Formulas MATH 41-PreCalculus Arc Length s = rθ Area of Sector A = 1 r θ Linear Speed v = s t, v = rω Angular Speed ω = θ t Acute Angle sin(θ) = b = opposite c hypotenuse csc(θ) = c = hypotenuse b opposite General Angle cos(θ) = a c = adjacent hypotenuse sec(θ) = c a = hypotenuse adjacent sin(θ) = b r cos(θ) = a r tan(θ) = b a csc(θ) = r,b 0 b sec(θ) = r,a 0 a cot(θ) = a,b 0 b Cofunctions tan(θ) = b = opposite a adjacent cot(θ) = a = adjacent b opposite sin(θ) = cos ( π θ), cos(θ) = sin (π θ), tan(θ) = cot (π θ) csc(θ) = sec ( π θ), sec(θ) = csc (π θ), cot(θ) = tan (π θ) Fundamental Identities tan(θ) = sin(θ) cos(θ) csc(θ) = 1 sin(θ) cot(θ) = cos(θ) sin(θ) sec(θ) = 1 cos(θ) cot(θ) = 1 tan(θ) sin (θ) + cos (θ) = 1 tan (θ) + 1 = sec (θ) cot (θ) + 1 = csc (θ) Even-Odd Identities sin( θ) = sin(θ) cos( θ) = cos(θ) tan( θ) = tan(θ) csc( θ) = csc(θ) sec( θ) = sec(θ) cot( θ) = cot(θ) Inverse Functions y = sin 1 (x) means x = sin (y) where 1 x 1 and π y π y = cos 1 (x) means x = cos (y) where 1 x 1 and 0 y π y = tan 1 (x) means x = tan (y) where x and π < y < π y = csc 1 (x) means x = csc (y) where x 1 and π y π, y 0 y = sec 1 (x) means x = sec (y) where x 1 and 0 y π, y π y = cot 1 (x) means x = cot (y) where x and 0 < y < π Auth:C.Villarreal-Prof. CBC 015Spring

6 MATH 41-PreCalculus Sum and Difference Formulas sin(α + β) = sin(α) cos(β) + cos(α) sin(β) sin(α β) = sin(α) cos(β) cos(α) sin(β) cos(α + β) = cos(α) cos(β) sin(α) sin(β) cos(α β) = cos(α) cos(β) + sin(α) sin(β) tan(α + β) = tan(α)+tan(β) 1 tan(α) tan(β) Half-Angle Formulas sin ( α ) = ± 1 cos(α) tan(α β) = tan(α) tan(β) 1+tan(α) tan(β) cos ( α ) = ± 1+cos(α) tan ( α ) = ± 1 cos(α) = 1 cos(α) 1+cos(α) sin(α) = sin(α) 1+cos(α) Double-Angle Formulas sin(θ) = sin(θ) cos(θ) cos(θ) = cos (θ) sin (θ) = cos (θ) 1 = 1 sin (θ) tan(θ) = tan(θ) 1 tan (θ) sin (θ) = 1 cos(θ) Product to Sum Formulas, cos (θ) = 1+cos(θ) sin(α) sin(β) = 1 [cos(α β) cos(α + β)] cos(α) cos(β) = 1 [cos(α β) + cos(α + β)] sin(α) cos(β) = 1 [sin(α + β) + sin(α β)], tan (θ) = 1 cos(θ) 1 cos(θ) Sum to Product Formulas sin(α) + sin(β) = sin ( α+β ) cos (α β ) sin(α) sin(β) = sin ( α β ) cos (α+β ) cos(α) + cos(β) = cos ( α+β ) cos (α β ) cos(α) cos(β) = sin ( α+β ) sin (α β ) Auth:C.Villarreal-Prof. CBC 015Spring

7 Law of Sines sin(a) a = sin(b) b = sin(c) c Law of Cosines a = b + c bc cos(a) b = a + c ac cos(b) c = a + b ab cos(c) Area of SSS Triangles (Heron s Formula) MATH 41-PreCalculus K = s(s a)(s b)(s c), where s = 1 (a + b + c) Area of SAS Triangles K = 1 ab sin (C), K = 1 bc sin (A), K = 1 ac sin (B) For y = Asin (ωx φ) or y = Acos (ωx φ), with ω > 0 Amplitude = A, Period= T = π ω, Phase shift = φ ω Auth:C.Villarreal-Prof. CBC 015Spring