Precalculus Review Functions to KNOW! 1. Polynomial Functions Types: General form Generic Graph and unique properties 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 Domain: x + 3 y = determine the following: x 5 Vertical Asymptote Horizontal or Slanted Asymptote Also helps to plot the intercepts: x-intercept y-intercept Ex) Determine the asymptotes (vertical, horizontal/slant) for 2 x x 6x + 7 a) f( x) = b) g( x) = 2 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: Range: Domain: 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: x y = a General Log. Function: y = log ( x) a Domain: Domain: Range: Range: Intercept: Intercept: Asymptote: Asymptote: Graph: Graph: Most frequently used base is whose log inverse is Approximate value:
Properties of Exponentials and Logarithms you ll need in calculus: Rewrite between exponential form and logarithmic form: x a = b can be rewritten as can be rewritten as ln( b) = x Cancellation Properties (VERY handy when solving equations) Solving equations: Solve the equation 3 1 10 x = 45 Solve the equation 6ln(15 7 x) + 20 = 38 Base Change Formula The Laws of Logarithms These are handy when you need to expand or condense logarithmic expressions. I. ln( UV ) = U II. ln( V ) = M III. ln( U ) =
5. Trigonometric Functions Trigonometric functions were defined in several ways: -Right Triangle Definitions: The main three and their reciprocals 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 ) =
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: 7π Ex) Evaluate csc( ) 4 Ex) Solve the equation 2 3tan ( x ) = 1 on the interval [0,2 π ). Ex) What interval (or intervals) make 2cost + 1 0 on the interval [0,2 π )? - 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 x- intercepts) Secant and Tangent have vertical asymptotes at every ODD multiple of 2 π (where Cosine has x- intercepts) Trigonometric Identities we will need in Calculus (KNOW THEM!) Reciprocal Identities sinu = 1 csc u cos u = 1 secu tan u = 1 cot u csc u = 1 sin u sec u = 1 cosu cot u = 1 tanu Pythagorean Identities also cos also tan also cot sin u + cos u = 1 u = 1 sin u and sin u = 1 cos 1+ tan u = sec u u = sec u 1 and 1 = sec u tan 1+ cot u = csc u u = csc u 1 and 1 = csc u cot u u u Quotient Identities sin u tan u = cosu cosu cot u = sinu Even / Odd Identities ODDS sin( u) = sin u csc( u) = cscu tan( u) = tanu cot( u) = cot u EVENS cos( u) = cosu sec( u) = secu