M1111 Final Exam Review PDF

Final Exam Review 2.1 The Rectangular Coordinate System and Graphs Graphing equations by plotting points Finding x- and y-intercepts Distance Formula: the distance between (𝑥𝑥1 , 𝑦𝑦1 ) and (𝑥𝑥2 , 𝑦𝑦2 ) is (𝑥𝑥1 − 𝑥𝑥2 )2 + (𝑦𝑦1 − 𝑦𝑦2 )2...

Final Exam Review 2.1 The Rectangular Coordinate System and Graphs Graphing equations by plotting points Finding x- and y-intercepts Distance Formula: the distance between (𝑥𝑥1 , 𝑦𝑦1 ) and (𝑥𝑥2 , 𝑦𝑦2 ) is (𝑥𝑥1 − 𝑥𝑥2 )2 + (𝑦𝑦1 − 𝑦𝑦2 )2 𝑥𝑥1 +𝑥𝑥2 𝑦𝑦1 +𝑦𝑦2 Midpoint Formula: the midpoint between (𝑥𝑥1 , 𝑦𝑦1 ) and (𝑥𝑥2 , 𝑦𝑦2 ) is , 2 2 2.2 Linear Equations Solve linear equations in one variable (Fractions? Multiply both sides by the LCD) 𝑦𝑦 −𝑦𝑦 Find equations for lines: slope 𝑚𝑚 = 𝑥𝑥2−𝑥𝑥1 (rise/run) 2 1 Point-slope form: 𝑦𝑦 − 𝑦𝑦1 = 𝑚𝑚(𝑥𝑥 − 𝑥𝑥1 ) Slope-intercept form: y = mx + b Slopes of parallel and perpendicular lines 2.3 Word Problems (5-step procedure) 1. Use a variable (x) for the quantity you are asked to find 2. Write other quantities in terms of x 3. Write an equation based on steps 1 and 2 4. Solve the equation 5. Answer the question in words 2.4 Complex numbers Define 𝑖𝑖 = √−1 (Notice that 𝑖𝑖 2 = −1.) A complex number is a + bi, where a and b are real numbers The complex conjugate of a + bi is a – bi. Add, subtract, multiply and divide complex numbers To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator. 1.5 Factoring Factor the Greatest Common Factor (GCF) Factor trinomials, including perfect squares Factor a difference of two squares: 𝑎𝑎2 − 𝑏𝑏 2 = (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎 + 𝑏𝑏) Factor the sum and difference of two cubes:  𝑎𝑎3 − 𝑏𝑏 3 = (𝑎𝑎 − 𝑏𝑏)(𝑎𝑎2 + 𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 ), 𝑎𝑎3 + 𝑏𝑏 3 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎2 − 𝑎𝑎𝑎𝑎 + 𝑏𝑏 2 ) Factor by grouping 2.5 Quadratic Equations Standard form: 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0 Factoring Method and the Zero Product Property: If 𝐴𝐴𝐴𝐴 = 0, then either A = 0 or B = 0. Square Root Property: If 𝑢𝑢2 = 𝑝𝑝, then 𝑢𝑢 = ± 𝑝𝑝 Completing the Square −𝑏𝑏±√𝑏𝑏 2 −4𝑎𝑎𝑎𝑎 Quadratic Formula: If 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0, then 𝑥𝑥 = 2𝑎𝑎 2.6 Other Types of Equations Equations with rational exponents Factoring to solve higher-order equations Equations with radicals (isolate the radical; then square both sides) Equations with absolute values: If |𝑢𝑢| = 𝑝𝑝, then u = p or u = - p 2.7 Linear and Absolute Value Inequalities Interval notation: [a, b] includes the endpoints at a and b; (a, b) does not include endpoints Properties of Inequalities Reverse the inequality if you multiply or divide both sides by a negative number Compound (“double”) inequalities: 𝑎𝑎 ≤ 𝑥𝑥 < 𝑏𝑏 means 𝑎𝑎 ≤ 𝑥𝑥 and 𝑥𝑥 < 𝑏𝑏 Abs. value inequalities: |𝑢𝑢| ≤ 𝑝𝑝 means −𝑝𝑝 ≤ 𝑢𝑢 ≤ 𝑝𝑝; while |𝑢𝑢| ≥ 𝑝𝑝 means 𝑢𝑢 ≤ −𝑝𝑝 or 𝑢𝑢 ≥ 𝑝𝑝 3.1 Functions and Function Notation Determine whether a relation is a function (Vertical Line Test) Find the value of a function from a table, a graph, or an algebraic rule Determine whether a function is one-to-one (Horizontal Line Test) Identify the Basic Toolkit Functions and their graphs 3.2 Domain and Range The domain is the largest possible set of inputs for which the function is defined. The range is the set of outputs (y-coordinates). For the domain, what’s under a square root should be greater than or equal to 0; what’s in the denominator should not be equal to 0. 3.3 Rates of Change and Behavior of Graphs 𝑓𝑓(𝑏𝑏)−𝑓𝑓(𝑎𝑎) The average rate of change of f (x) over [a, b] is 𝑏𝑏−𝑎𝑎. Determining where a function is increasing and decreasing  Local maximum and minimum values Absolute maximum and minimum values 3.4 Combinations of Functions Given two functions f and g, we can combine them using algebraic operations: (𝑓𝑓 + 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥), (𝑓𝑓 − 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) − 𝑔𝑔(𝑥𝑥), 𝑓𝑓 𝑓𝑓(𝑥𝑥) (𝑓𝑓𝑓𝑓)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥)𝑔𝑔(𝑥𝑥), (𝑥𝑥) = 𝑔𝑔 𝑔𝑔(𝑥𝑥) We can also compose the two functions: (𝑓𝑓°𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑔𝑔(𝑥𝑥)) and (𝑔𝑔°𝑓𝑓)(𝑥𝑥) = 𝑔𝑔(𝑓𝑓(𝑥𝑥)) 3.5 Transformations Vertical and Horizontal Shifts Vertical and horizontal stretches and compressions (squishes) Reflections about the x- and y-axes Changes in the output correspond to vertical changes (normal) Changes in the input correspond to horizontal changes (backwards) Even functions: 𝑓𝑓(−𝑥𝑥) = 𝑓𝑓(𝑥𝑥) for all x in the domain of f (symmetry about y-axis) Odd functions: : 𝑓𝑓(−𝑥𝑥) = −𝑓𝑓(𝑥𝑥) for all x in the domain of f (symmetry about the origin) 3.6 Absolute Value Functions 𝑦𝑦 = |𝑥𝑥| has V-shape, with minimum point at the origin Know transformations with |x| 3.7 Inverse Functions f and g are inverse functions if 𝑓𝑓 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 = 𝑔𝑔(𝑓𝑓(𝑥𝑥)) for all x in the domains of f and g 3-step procedure for finding an inverse function 1. Set y = f (x). 2. Solve for x in terms of y. 3. Switch x’s and y’s and set 𝑓𝑓 −1 (𝑥𝑥) equal to what is equal to y. One-to-one functions have inverse functions Restrict the domain of a function to make it one-to-one; then find its inverse function The graphs of inverse functions are reflections of each other about the line y = x. 5.1 Quadratic Functions  General form: 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐; opens up if a > 0; opens down if a < 0 Standard form: 𝑓𝑓(𝑥𝑥) = 𝑎𝑎(𝑥𝑥 − ℎ)2 + 𝑘𝑘 ; vertex (h, k), axis of symmetry x = h Complete the square to get from general form to standard form −𝑏𝑏 Shortcut: ℎ = 2𝑎𝑎 , k = f (h) Maximizing or minimizing a quadratic function: find the vertex 5.2 Power Functions and Polynomials Identify end behavior of polynomials based on degree (even/odd), leading coefficient (+/-) A polynomial of degree n has at most n x-intercepts and at most (n – 1) turning points 5.3 Graphs of Polynomial Functions Correspondence between factors (x – a) and zeros (x = a) of a polynomial Multiplicity of zeros and their effect on the graph of the polynomial Graph polynomials with end behavior, zeros with correct multiplicities, y-intercept 5.4 Dividing Polynomials Long Division Synthetic Division when dividing by x – a 5.6 Rational Functions Domain Vertical Asymptote(s): Zero(s) of the denominator* Horizontal Asymptote (Ratio of leading terms; end behavior) Slant Asymptote (when degree of numerator = 1 + degree of denominator) Do long division; the slant asymptote will be y = quotient x-intercept(s): Zero(s) of the numerator* y-intercept: y = f (0) *Watch out for holes; where the same factor appears in both the numerator and denominator 4.2 Modeling with Linear Functions Use the given information to identify two points and find the slope m of the line Use point-slope form 𝑦𝑦 − 𝑦𝑦1 = 𝑚𝑚(𝑥𝑥 − 𝑥𝑥1 ) to find the equation (maybe w/ other variables) 6.1 Exponential Functions Exponential growth and decay Compound interest (compounded n times per year and compounded continuously) Using a calculator to evaluate exponential functions (with base e, and other bases) 6.2 Graphs of Exponential Functions  Identify the domain, range and (horizontal) asymptote Transformations of exponential functions: keep track of domain, range, and asymptote 6.3 Logarithmic Functions Logarithmic functions are inverses of exponential functions Convert from logarithmic form (log 𝑎𝑎 𝑥𝑥 = 𝑦𝑦) to exponential form (𝑎𝑎 𝑦𝑦 = 𝑥𝑥) and vice versa Evaluate logarithms, including common logarithms and natural logarithms 6.4 Graphs of Logarithmic Functions The graph of 𝑦𝑦 = log 𝑎𝑎 𝑥𝑥 is the reflection of the graph of 𝑦𝑦 = 𝑎𝑎 𝑥𝑥 about the line y = x Identify the domain, range, and (vertical) asymptote Transformations of logarithmic functions: keep track of domain, range, and asymptote 6.5 Properties of Logarithms Product Rule: log 𝑎𝑎 (𝐴𝐴𝐴𝐴) = log 𝑎𝑎 (𝐴𝐴) + log 𝑎𝑎 (𝐵𝐵) 𝐴𝐴 Quotient Rule: log 𝑎𝑎 𝐵𝐵 = log 𝑎𝑎 (𝐴𝐴) − log 𝑎𝑎 (𝐵𝐵) Power Rule: log 𝑎𝑎 (𝐴𝐴𝐵𝐵 ) = 𝐵𝐵 log 𝑎𝑎 (𝐴𝐴) log 𝑥𝑥 ln 𝑥𝑥 Change of Base Formula: log 𝑎𝑎 (𝑥𝑥) = log 𝑎𝑎 = ln 𝑎𝑎 6.6 Solving Exponential and Log Equations Solving Exponential Equations 1. Isolate the exponential expression 2. Convert to log form (or take the log of both sides) and solve for x Solving Log Equations 1. Consolidate multiple logs into a single log (if necessary) 2. Isolate the logarithmic expression 3. Convert into exponential form to solve for x 6.7 Exponential and Log Models Continuous growth rate: 𝐴𝐴 = 𝐴𝐴0 𝑒𝑒 𝑟𝑟𝑟𝑟 , where r is the continuous growth rate (as a decimal) Exponential decay: 𝐴𝐴 = 𝐴𝐴0 𝑒𝑒 −𝑟𝑟𝑟𝑟 , where r is the decay rate 1 If h is the half life of a substance, 𝐴𝐴 = 𝐴𝐴0 (2)𝑡𝑡/ℎ