Summer Assignment for AP Preca lc ulus 2023 2034 Name ________________________________ This assignment is a review Algebra 2 and a lgebraic concepts that you need to be familiar with in order to make a smooth transition into AP Precalculus . It will be due when you return to school NO EXCEPTIONS. A test over this material will be given during the 1st week of school The AP Prec alculus team looks forward to meeting you and working with you . In the m eanti me, have a great summer! ▪ For help on Algebra 2 conce pts: http://www.khanacademy.org/ Scroll down on the home page to see many topics of interest. I. Analyzing Functions Use the figure at the right to answer the problems below: 1. Use interval notation to determine the interval on which the function, ( ) f x , is Increasing: _____________ Decrea sing:___________ 2. The zero s of the fun ction are ___________ x = 3. For what values of x is ( ) f x =0? 4 . Find the following information about the given graph. Be careful when to use x - values and y - values! a. Domain __________ b. Range ____________ c. Intervals of increase_______________ d. Intervals of decrease ______________ e. Relative (local) maximum ____________ f. Relative (local) minimum ____________ g. Absolute maximum ______________ i. x - intercepts _____________________ j. y - intercept ______________________ k. Even, odd, or neither ______________ l. End behavior: ________________ Michael Sutu (0, 1) [-1, 0) [1,2] [1,2] All real numbers All real numbers (-inf,-1.12) and (1.786,inf) (-1.12,1.786) (-1.12, 2.477) (1.786, -5.007) (-2,0)(0,0) and (3,0) (0,0) As x approaches -inf, y approaches -inf As x approaches inf, y approaches inf Odd None(inf) h. Absolute minimum _None(-inf)__________ Constant: ___ [ 1, 2 ] _________ Concave Up:____ [ -1,0)(0,1)____ Concave Down: ___None_____ Summer Assignment for AP Preca lc ulus 2023 2034 Name ________________________________ 5. Let 2 ( ) 3 f x x = and 9 ( ) , 1, 1 x g x x x − = − + find the following. State doma in restrictions when necessary. a. ( ) ( ) _____ f g x = b. ( ) ( ) ____________ g f x = c. ( ) 1 __________ f x − = d. Domain, range and zeros of ( ) f x e. Domain, range and zeros of ( ) g x Find 1 ( ) f x − then verify that it is the inverse by doing ( ) ( ) 1 f f x x − = and ( ) ( ) 1 f f x x − = Your work for the compositions SHOULD NOT BE INDENTICAL! 6 ( ) 2 3 f x x = + 7 3 ( ) 1 f x x = − Graph the piece - wise function, then evaluate the function at the values indicated. 8 3 2, 3 ( ) 4, 3 x x f x x x + = − + 9 ( ) 2 2 1, 2 4, 2 1 3 1, 1 3 1, 3 x x x f x x x x x − − − = + − ( ) ( ) ( ) ( ) 2 _____ ( ) 3 _____ ( ) 5 _____ a f b f c f = = = ( ) ( ) ( ) ( ) ( ) 5 _____ ( ) 3 _____ ( ) 2 _____ ( ) 0 _____ a f b f c f d f = − = = = (3x^2-9)/(3x^2+1) +- sqrt x/3 Domain: All Real #'s Range: x >= 0 Zeros: (0, 0) Domain: All Real except for x = -1 Range: All real except for y = -1 Zeros: (9, 0) Inverse f(x) = (3 - x)/2 x = 2y + 3 0 = 2y + 3 - x 2y = 3 - x y = (3 - x)/2 Inverse of f(x) = cube rt -1 - x x = y^3 - 1 y^3 = -1 - x y = cube rt -1 -x 2 1 17 24 8 7 4 (3x^2-54x+243) /(x^2+2x+1 x>=0 Summer Assignment for AP Preca lc ulus 2023 2034 Name ________________________________ Determine which functions are odd, even or neither. Be able to explain both graphically and algebraically. 10. 2 ( ) 2 7 f x x = − 11. 3 ( ) 4 2 f x x x = − − 12. 2 ( ) 4 4 4 f x x x = − + Find the domain of each function. Recall: The denominator of a rational function can never = 0 and any radical with an even index can’t have a negative radicand. 1 3 3 2 4 1 ( ) x x f x − + = 1 4 2 4 2 4 ( ) x x g x − + = 1 5 2 2 5 6 3 18 ( ) x x x x h x − − − − = 16 2 9 2 9 x x y − + = Asymptotes: State the equation for the vertical and horizontal asymptotes for the following functions. Find the coordinates of any holes the function has. 17 3 ( ) x x f x − = 18 2 4 1 ( ) x x g x + − = 19 2 3 2 9 3 18 ( ) x x x x f x − − − = II. Factoring, simplifying, and solving equations Factor each 2 0 4 2 11 80 x x + − 2 1 3 2 2 3 x xy x y y − + − Solve each of the following for x. 22 2 9 0 x − = 23 2 3 7 3 0 x x + + = 24 2 5 24 0 x x − − = 25 2 5 3 1 0 x x + + = 26 ln 3 x = 27 ln 1 x = 28 ln x e = Even Odd Neither All real numbers except for x = -0.25 All real numbers except x >= 4.5 x = 3 x = 1 x = -1 y = 0 x = 0 x = 6 All real numbers excluding -2 for x = -3 and 6 (x^2 - 5)(x^2 + 16) (x + y)(x + y)(x - y) x = +-3 x=-+(7-sqrt13)/(6) x=8,-3 x=-5/2 x=e^3 x=e x = e^e Summer Assignment for AP Preca lc ulus 2023 2034 Name ________________________________ I II Logarithms and Exponentials Evaluate each logarithm or exponential WITHOUT u sing a calculator. 2 9. 2 log 8 ____ = 3 0. 1 4 64 log _____ = 3 1. log10 ______ = 3 2. 5 log 7 5 ______ = 3 3. ln ____ e = 3 4. log1 ____ = 3 5. ln1 _____ = 3 6. 5 log 1 _____ = Solve each equation ALGEBRAICALLY. 37 ( ) 2 log 3 1 ________ x x x − = = 38 ( ) ln 4 0 ________ x x + = = 39 3 1 5 625 ________ x x + = = 40 2 1 3 30 ______ x x − = = I V. Lines Find the equation of the line going through each set of points. Use the Point - Slope Formula: ( ) 1 1 y y m x x − = − Recall: 2 1 2 1 y y x x m − − = 41 ( ) ( ) 2,3 , 5,6 42 ( ) ( ) 12,1 , 5,0 43 ( ) ( ) 3, 7 , 3,14 − − 44 ( ) ( ) 5,5 , 8,5 Write the equation of a line that fits the required directions: 45 . The equation of a line parallel to # 41 going through (1,1). 46 . The equation of a line perpendicular to # 4 2 going through (3,5). 47 . The equation of a line parallel to # 4 3 going through (2,4) 48 . The equation of a line perpendicular to # 4 4 going through the point ( - 5,2). y = x + 1 y = 1/7x - 5/7 x = -3 y = 5 y = x y = 7/1x - 16 x = 2 x = -5 3 -3 1 7 1 0 0 0 5, -2 -3 1 1/2, (ln(30))/(2ln(3)) Summer Assignment for AP Preca lc ulus 2023 2034 Name ________________________________ V .PA RENT FUNCTIONS 49 . Graph each of the parent functions and be familiar with them enough to graph them later WITHOUT the use of a calculator. f. ( ) f x x = Summer Assignment for AP Preca lc ulus 2023 2034 Name ________________________________ All real All real (0,0) (0,0) (-inf,inf) None (0,inf) (-inf,0) None None All real x >= 0 (0,0) (0,0) (0,inf) (-inf,0) (-inf,inf) None None None x >= 0 x >=0 (0,0) (0,0) (0,inf) None (0,inf) None None None All real y = # None (0, #) None None (-inf,inf) None None None All real All read (0, 0) (0, 0) (-inf, inf) None (0, inf) (-inf, 0) None None All real All real (0, 0) (0, 0) (-inf, inf) None (0, inf) (-inf, 0) None None All real except for 0 All real except for 0 None None None (-inf, 0)(0, inf) (0, inf) (inf, 0) x = 0 y = 0 All real x >= 0 (0, 0) (0, 0) (0, inf) (-inf, 0) (-inf, inf) None None None Summer Assignment for AP Preca lc ulus 2023 2034 Name ________________________________ All real (0, inf) None (0, 1) (-inf, inf) None (-inf, inf) None None None All real y = 0.5 None (0, 0.5) None None (-inf, inf) None None None x > 0 All real (1, 0) None (0, inf) None (1, inf) (0, 1) x = 0 None x > 0 All real (1, 0) None None (0, inf) (0, 1) (1, inf) x = 0 None