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Precalculus Review Functions to KNOW! 1. Polynomial Functions Types: General form and unique properties Generic Graph Constants Linear Quadratic Cubic Generalizations for Polynomial Functions - The domain for all polynomial functions is ALWAYS _______________ - The degree of a polynomial is ________________________________ - Graph behavior based on its degree: End behavior: Number of ‘turning points’ A function is said to be even if A function is said to be odd if 2. Rational Functions (a.k.a. Fractional Functions) - General form: - Domain: - Unique Graph attributes: Vertical asymptotes: Horizontal or Slanted Asymptotes: 1. degree numerator < degree denominator 2. degree numerator = degree denominator 3. degree numerator > degree denominator Ex) For the function y = x +3 determine the following: x −5 Domain: Vertical Asymptote Horizontal or Slanted Asymptote Also helps to plot the intercepts: x-intercept y-intercept Ex) Determine the asymptotes (vertical, horizontal/slant) for … x2 − 6x + 7 x a) f (x) = 2 b) g(x) = x −9 x −5 3. Inverse Functions In order for a function f (x) to have an inverse, it must be ______________________ … which means The inverse of the function f (x) is denoted as ________________ Ex) f (x) = x 3 Ex) h(x) = x 2 The purpose of an inverse function is to ______________________________________ Properties of inverse functions: - Domain and Range: - Graph symmetry Ex) Sketch the graph of the inverse of the function f (x) on the blank axes. Domain: Domain: Range: Range: 4. Exponential and Logarithmic Functions Exponential functions of base a and Logarithmic functions of base a are inverses of each other. General Exp. Function: y = a x General Log. Function: y = log a ( x) Domain: Domain: Range: Range: Intercept: Intercept: Asymptote: Asymptote: Graph: Graph: Most frequently used base is ______ Approximate value: … whose log inverse is ______________ Properties of Exponentials and Logarithms you’ll need in calculus: Rewrite between exponential form and logarithmic form: a x = b can be rewritten as _______________ __________ can be rewritten as ln(b) = x Cancellation Properties (VERY handy when solving equations) Solving equations: Solve the equation 103 x−1 = 45 Solve the equation 6ln(15 − 7x) + 20 = 38 Base Change Formula The Laws of Logarithms These are handy when you need to expand or condense logarithmic expressions. I. ln(UV ) = II. ln( UV ) = III. ln(UM ) = 5. Trigonometric Functions Trigonometric functions were defined in several ways: -Right Triangle Definitions: The main three … sinθ = cosθ = tanθ = REMEMBER: The roles of ‘OPPOSITE’ and ‘ADJACENT’ depend on which acute angle you’re calling θ . - Unit Circle Definitions: Let t be a radian angle measure and ( x , y) represents the point on the unit circle paired with the angle t sin(t ) = cos(t ) = tan(t ) = and their reciprocals FOR REFERENCE ONLY!!! THIS IS PREREQUISITE MATERIAL!!! YOU WILL NOT BE ALLOWED TO USE THIS ON THE TEST!!! You’ll need the unit circle for various reasons this semester: Ex) Evaluate csc( 74π ) Ex) Solve the equation 3tan2 (x) = 1 on the interval [0,2π ) . Ex) What interval (or intervals) make 2cos t + 1 ≤ 0 on the interval [0,2π ) ? - Trigonometric Function Graphs y = sin(x) y = cos(x) Domain of Sine and Cosine: ____________ Range of Sine and Cosine: _____________ Graph is periodic with a cycle repeating every interval of length ______________. Ex) Sketch two full periods of the graph of y = −8cos(10 x) . How does the –8 affect the graph? How does the 10 affect the graph? The other trigonometric function graphs for reference (again … prerequisite material won’t be given on the test) Graph of y = csc(x) Graph of y = sec(x) Graph of y = tan( x) Graph of y = cot(x) y = tan x y = cot x Cosecant and Cotangent have vertical asymptotes at every multiple of π (where Sine has xintercepts) Secant and Tangent have vertical asymptotes at every ODD multiple of π2 (where Cosine has xintercepts) Trigonometric Identities we will need in Calculus (KNOW THEM!) Reciprocal Identities 1 sin u = csc u 1 csc u = sin u 1 cos u = sec u 1 sec u = cos u 1 tan u = cot u 1 cot u = tan u Quotient Identities sin u cos u cos u cot u = sin u tan u = Pythagorean Identities Even / Odd Identities ODDS sin 2 u + cos2 u = 1 also … cos2 u = 1 − sin 2 u and sin 2 u = 1 − cos2 u sin( − u) = − sin u csc( − u) = − csc u tan( − u) = − tan u cot( − u) = − cot u 1 + tan 2 u = sec 2 u also … tan 2 u = sec2 u − 1 and 1 = sec 2 u − tan 2 u EVENS 2 2 1 + cot u = csc u also … cot u = csc 2 u − 1 and 1 = csc 2 u − cot 2 u 2 cos( − u) = cos u sec( − u) = sec u