Exponent Rules (Chapter 3) Important Math 125 Definitions/Formulas/Properties Let m & n be integers and a & b real numbers. Product Property Quotient Property Power to a Power Product to a Power Quotient to a Power * + a % a & = a %(& a % a & = a%)& a % & = a %& ab & = a & b & & = * for b 0 + Zero Exponent a / = 1 for a 0 Negative Exponent a )& = 1 * & 1 * 2 = a& for a 0 Multiplying/Factoring Binomials (Chapters 3 & 4) Square of a Binomial/ Perfect Square Trinomial a ± b 4 = a 4 ± 2ab + b 4 Pythagorean Theorem (Chapter 4) For a right (90 ) triangle: Multiplying Conjugates/ Difference of Squares a + b a b = a 4 b 4 Sum/Difference of Cubes a 8 ± b 8 = (a ± b)(a 4 ab + b 4 )* *The trinomial a 4 ab + b 4 is prime and cannot be factored any further a 4 + b 4 = c 4 Formulas Relating Distance Rate & Time (Chapter 4) Formula For Combined Work (Chapter 4) d = rt t = d r Let t 1 be the time it takes for the first person or group to complete a particular task. Let t 4 be the time it takes for the second person or group to complete this task. If t is the time it takes when they work together on the task then 1 t 1 + 1 t 4 = 1 t
Linear Equations (Chapter 6) Slope Formula m = y 4 y 1 x 4 x 1 Parallel Slope m 1 = m 4 Perpendicular Slope m 1 = 1 m 4 Slope-Intercept Form* y = mx + b Point-Slope Form y y 1 = m(x x 1 ) Standard Form Horizontal Line Vertical Line ax + by = c x = k y = k Properties of Radicals (Chapter 8) *In function notation we can write slope-intercept form as f(x) = mx + b. Let m & n be positive integers where n > 1 and a & b be non-negative real numbers. N th Root of a Perfect Power a & = a Rational Exponents a %/& = a % or % a Product Rule a b = ab Quotient Rule * = * + + for b 0 Distance & Midpoint Formulas (Chapter 8) Square Root Property (Chapter 9) Distance Formula d = x 4 x 1 4 + y 4 y 1 4 Midpoint Formula M = x 1 + x 4 2 y 1 + y 4 2 Isolate the square term to one side of the equation then apply the square root to both sides: x 4 = b x 4 = b x = ± b ax + c 4 = b ax + c 4 = b ax + c = ± b Quadratic Formula (Chapter 9) If ax 4 + bx + c = 0 for a 0 then x = b ± b4 4ac 2a
Parabolas in Standard Form (Chapter 9) Standard Form f x = ax 4 + bx + c a 0 Axis of Symmetry Vertex y-intercept x = b 2a b b f 2a 2a Evaluate f 0 = c x-intercept(s) Solve f x = 0 We us all the above information to graph a parabola. Parabolas in Vertex Form (Chapter 9) Let h & k be real numbers. Vertex Form f x = a(x h) 4 + k a 0 Vertex Steps to Graph (h k) 1. Graph y = ax 4 2. Move y = ax 4 h units horizontally & k units vertically. Graphs of Parent Functions (Chapter 10) Graphing Functions using Shifting & Reflecting (Chapter 10) We generalize graphing functions based using shifting and reflectioings the above parent functions.
Generalizing Trandfomations: y = af(x h) + k To generalize we let h & k be real numbers (like vertex form of a parabola). Ø Identify the parent function f(x) and make a table of its key points. Ø Graph y = af(x). o o If a = 1 then the graph opens upward. If a = 1 then the graph opens downward. Ø Shift the graph of y = af(x) horizontally by h units then vertically by k units. Ø The origin of f x is now shifted to its new center (h k). o This point is a vertex if the parent function is a square or absolute value function. Variation (Chapter 10) We create new variations using the 3 properties above.
Absolute Value Equations (Chapter 11) Isolate the absolute value term to one side then apply one of the four properties below: Equations Involving Absolute Values Let a is a positive real number and x be any algebraic expression. If x = a then x = a or x = a. If x = 0 then x = 0. If x = a then x = (no solution; empty set) If x = y then x = y or x = y Absolute Value Inequalities (Chapter 11) Isolate the absolute value term to one side then apply one of the two properties below: Algebra of Functions (Chapter 12) Inequalities Involving Absolute Values Let a is a positive real number and x be any algebraic expression. If x < a then a < x < a o < can be interchanged with. If x > a then x < a or x > a o > can be interchanged with. Function Operations Let f(x) and g(x) be defined then f + g x = f x + g(x) f g x = f x g x f g x = f x g(x) Y Z x = Y [ Z([) Function Composition (Chapter 12) for g(x) 0 Let f(x) g x & f(g x ) be defined then One-to-One Functions (Chapter 12) f g x = f(g x ) A function is one-to-one if each y corresponds to one x. Ø The above tells us a one-to-one function never has repeating y-values. The Horizontal Line Test If every horizontal line intersects the graph of a function at most once then the function is one-to-one.
Inverse Functions (Chapter 12) The Inverse Function Let f(x) be a one-to-one function. Then If f x = y then x = f )1 (y). Finding & Verifying Functions (Chapter 12) o This tells us that if the point x y is on the graph of f then y x is on the graph of f )1. That is to say the graph of f )1 is found by reflecting the points f about the line y = x. Steps to Find f )1 (x) Let f(x) be a one-to-one function. 1. Replace f x with y. 2. Interchange x and y. 3. Solve for y. 4. Replace y with f(x). Logarithms (Chapter 12) Verifying Inverses Two one-to-one functions f(x) and g x are inverses if and only if f g x = x & g f x = x. For x > 0 b > 0 and b 1 the logarithm function of base b of x if defined by where b d = x. Exponential Form b d = x y = log + x Logarithmic Form y = log + x To solve a logarithmic equation we make sure it is written in logarithmic form the convert it to exponential form. ie log e (x + 1) = 1 4 9g h = x + 1 x + 1 = 9 x + 1 = 3 x = 2 Properties of Logarithms (Chapter 12) Properties of Logarithms For any real numbers x y & b > 0 with b 1 and any real number r: 1. Product Rule: log + x y = log + x + log + y [ 2. Quotient Rule: log = log + d + y log + y 3. Power Rule: log + x j = r log + x Note: If we use Properties (1-3) in solving a logarithmic equation we MUST check our solutions.
4. Special Properties: a. log + 1 = 0 b. log + b = 1 c. log + b [ = x d. b klm n[ = x 5. Change of Base: log + x = klm [ klm + or kp [ kp + Common Logarithm log 1/ x = log x log 10 [ = x 10 klm [ = x Natural logarithm log q x = ln x ln e [ = x e kp [ = x Solving Exponential Equations (Chapter 12) Ø If the both sides of the exponential equation can be written in terms of the same base we use the property below: Uniqueness of b x For any real numbers x y & b with b > 0 & b 1 if b [ = b d then x = y. ie 2 [ = 1 t 2[ = 1 4 u 2[ = 2 )8 x = 3 27 [ 3 [(1 = 9 4 3 8 [ 3 [(1 = 3 4 4 3 8[ 3 [(1 = 3 w 3 w[(1 = 3 w 4x + 1 = 4 x = 3 4 Ø If the both sides of the exponential equation cannot be written in terms of the same base we use the property below: Logarithmic Property of Equality For any positive real numbers x y & b with b 1 1 if x = y then log + x = log + y. ie 2 [ = 5 log 2 [ = log 5 x log 2 = log 5 x = klm y 8 e[(1 = 5 e [(1 = 15 ln e [(1 = ln 15 x + 1 = ln 15 x = 1 + ln 15. klm 4 Disclaimer: The formula sheet is meant as a study aid and not a comprehensive list or review of all topics in this course. You are still required to look over old exams quizzes homework classwork and notes for all we have covered this semester. Furthermore these formulas need to be committed to memory to successfully complete problems on the Final Exam.