Essential Algebra 2 Formula Cheat Sheet, Cheat Sheet of Algebra

Download Essential Algebra 2 Formula Cheat Sheet and more Cheat Sheet Algebra in PDF only on Docsity! Algebra 2 Formulas Essential Formulas for Algebra 2 Final Exam Laws of Exponents Multiply Powers of the Same Base = Adding Exponents (a’"\(a") =a" “n m Divide Powers of the Same Base = Subtracting Exponents fo _ a" a Power Rule = Multiplying Exponents (a"y" =q"*" Zero Exponent = 1 a =1 (aby" =a’ Distribution of Exponent with Multiple Bases a\" a” b) bt nd one Negative Exponent = Reciprocal a" pb" a = 1 —n —ny-n (ab) =a b- = ab" Distribution of Negative Exponent with Multiple Bases bY (5 )- (%) -3 atb#vVat+~vb Vaxb=Vaxvb a—b#Va-~b Va+b = Va=Vb Properties of Radicals ab = (fa)5) Distribution of Radicals of the Same Index (where a > 0 and b = 0 if n is even) Je — va b Xb Power Rule of Radicals = Multiplying Exponents winlq = (ma la" =a (ifn is odd) Reverse Operations of Radicals and Exponents Va" =|a| (if 7 is even) Page 1 of 10. Algebra 2 Formulas m nlm The index of the radical is the a"=va denominator of the fractional exponent. Special Products (A+B)(A -B)=4°- B’ (A+ BY =47+24B +B? (A+ By = 4° +34°B + 34B’ + B® (A- By = 4?-24B+ B? (A -B)’= 43 -347B + 34B’ - BS Special Expressions Difference of Squares AB = (A+ BYA-B) Perfect Trinomial Squares A’ +24B+ B= (A+ By Perfect Trinomial Squares 4” — 2AB + B?=(A- BY Sum of Cubes A’+ B*=(4+B)(4?— AB +B’) Difference of Cubes A’— B= (A-B)(4’7+ AB +B —b+vb? —4ac Quadratic Formula: x = 2a Discriminant = 5? — 4ac When Discriminant is Positive, b*- 4ac > 0 — Two Distinct Real Roots When Discriminant is Zero, b” - 4ac = 0 —> One Distinct Real Root (or Two Equal Real Roots) When Discriminant is Negative, Bb? — 4ac <0 — No Real Roots Note the pattern: , Product of Conjugate Complex Numbers ao, 2] Bo; 4 isi f=-1 P=-i =1 2 422-2 = 5. 6 7H. BL (a+ bi(a- bi =a - bv =a -b(-1) Psi P=-1 i=-i P=1 2 P=i P= 1 (a+ bi(a— bi) = a +B Pattern repeats every 4 power of i. Midpoint ofa LineSegment Distance of a Line Segement Slope x, +X, + V2 > 2 5-9, u=|2 Ate d= Y(x.—5,P +(,- 9) m= 22 2 2 x,-% Standard Equation for Circles @-hY+~-hW=r P (x, y) = any point on the path of the circle C(h, k) = centre of the circle r = length of the radius Page 2 of 10. Algebra 2 Formulas grxy=afxtht+k h = amount of horizontal movement h > 0 (move left); 1 < 0 (move right) k = amount of vertical movement k > 0 (move up); k < 0 (move down) Reflection off the x-axis Reflection off the y-axis 8) = ~fix) 8x) =f) All values of y has to switch signs but All values of x has to switch signs but all values of x remain unchanged. all values of y remain unchanged. Vertical Stretching and Shrinking Horizontal Stretching and Shrinking 8X) = af) 8(x) = flax) a is the Vertical Stretch Factor b is the Horizontal Stretch Factor a> 1 (Stretches Vertically by a factor of a) 0 < b <1 (Stretches Horizontally by a factor of 1/5) 0< a <1 (Shrinks Vertically by a factor of a) b> 1 (Shrinks Horizontally by a factor of 1/b) For Quadratic Functions in Standard Form of f(x) = a(x — hy +k Vertex at (h, k) Axis of Symmetry at x = h Domain: x ¢ R a= Vertical Stretch Factor a>0 Vertex at Minimum (Parabola opens UP) Range: y 2k (Minimum) a<0 Vertex at Maximum (Parabola opens DOWN) Range: y<k (Maximum) |a|>1 Stretched out Vertically |a|<1 Shrunken in Vertically h = Horizontal Translation (Note the standard form has x — / in the bracket!) h>0o Translated Right h<0 Translated Left k= Vertical Translation k>0 Translated Up k<0 Translated Down For Quadratic Functions in General Form: f(x) = ax” + bx + c y-intercept at (0, c) by lettingx = 0 (Note: Complete the Square to change to Standard Form) 2 x-intercepts at Game if b’ - 4ac > 0. No x-intercepts when b” - 4ac < 0 a Vertex locates at x = — = y= fi (- x) Minimum when a> 0; Maximum when a< 0 a a Sx) = One-to-One Function St te) = Inverse Function ») Ox) Domain of f(x) + Range of /'(x) Range of f(x) > Domain of f'@) Note: f(x) # — (Inverse is DIFFERENT than Reciprocal) F(x) Page 5 of 10. Algebra 2 Formulas End Behaviours and Leading Terms T L....\ a L “ey i Odd Degree Odd Degree Even Degree Even Degree Polynomial Function Polynomial Function Polynomial Function Polynomial Function and Positive Leading and Negative Leading and Positive Leading and Negative Leading Coefficient, a>0 Coefficient, a< 0 Coefficient, a>0 Coefficient, a< 0 Odd Degree Polynomial Functions When a > 0, Left is Downward (y — —00 as x > -00) and Right is Upward (y > © asx + ~). When a < 0, Left is Upward (y — © asx —> -0) and Right is Downward (y — -«© asx — ©). Even Degree Polynomial Functions When a > 0, Left is Upward (y — © asx — -c) and Right is Upward (y > © asx + ~), When a< 0, Left is Downward (y — -00 as x —> -0) and Right is Downward (y —> -00 asx —> 0). Multiplicity: - when a factored polynomial expression has exponents on the factor that is greater than 1. y y Even Multiplicity or Multiplicities of 2, 4, 6... means x-intercept is tangent to the x-axis x Odd Multiplicity or Multiplicities of 3, 5, 7... means inflection at x-int P(x) = ax(x — bY 8° Ge OP AO P(x) = ale - BY 7 “= ©) Polynomial Function Divisor Function In general, for P(x) + D(), we can write P(x) R —— = QO(x)+—-_ or P(x) = Dx) Q@&) +R Dw) Ax) De (x) = DE a Restriction: D(x) 4 0 Quotient Function Remainder Page 6 of 10. Algebra 2 Formulas If R = 0 when a , then (x — D) is a factor of P(x) and P(b) = 0. x P(x) = Dx) x O@%) P(x) = Original Polynomial D(x) = Divisor (Factor) Q(x) = Quotient If R 4 0 when ae , then (x - b) is NOT a factor of P(x). x P(x) = DX) x QO) + RO) The Remainder Theorem: To find the remainder of 20), Substitute 5 from the Divisor, (x — 5), into the Polynomial, P(x). x- In general, when re , P(b) = Remainder. x- To find the remainder of fe : Substitute (2) from the Divisor, (av — 5), into the Polynomial, P(x). a P(x) , P(#)= Remainder. ax In general, when The Factor Theorem: 1. If Pe) gives a Remainder of 0, then (x —d) is the Factor of P(x). x- OR If P(b) = 0, then (x — 5) is the Factor of P(x). 2. If 2e) gives a Remainder of 0, then (av —5) is the Factor of P(x). a OR If P(2) = 0, then (ax — b) is the Factor of P(x). Rational Roots Theorem: For a polynomial P(x), a List of POTENTIAL Rational Roots can be generated b: ALL the Factors of its Constant Term by ALL the Factors of its Leading Coefficient. ALL Factors of the Constant Term ALL Factors of the Leading Coefficient Potential Rational Zeros of P(x) = The Zero Theorem There are 7 number of solutions (complex, real or both) for any n™ degree polynomial function accounting that that a zero with multiplicity of k is counted & times. Page 7 of 10.