st Semester Pre Calculus Exam Review You will not receive hints on your exam. Make certain you know how to answer each of the following questions. This is a test grade. Your WORK and EXPLANATIONS are graded since the key is provided.. For the data given use a graphing utility to find the type of equation (bonus if you can find the line of best fit!. Write the steps below x 5 6 8 y 6 7 6 9. Find the x intercept(s) of: a. 5x y = b. y = (x ) 7 c. y = x 0x using the quadratic formula d. y = x 0x by solving for y, factoring and setting the factors equal to zero (because y=0 at the x intercept) e. How can you check your answer?. Find the y intercept(s) of: a. 5x y = b. y = (x ) 9 c. y = x 0x d. What are the steps for finding the y intercepts algebraically?(hint: what is equal to 0?). Find the vertex of: a. y = (x ) 9 Describe how you found your answer (Hint: what form is it in?) b. y = x 0x using part of the quadratic formula c. y = x 0x by completing the perfect square 5. Are the following functions even, odd, or neither? Explain your answer. a. y = x x b. y = x x c. d. 6. Solve the following algebraically, then graph the solutions on a number line: a. 7(x 8 ) = 7(6 x ) 8 b. 5b 6 b 9 c. x x
7. Complete the following by factoring: x x 6 = 0 8. Find f( ) given that a. f (x) = x x x 0 x b. f (x) = x 9. Given f (x) = x and g (x) = x find each of the following: a. ( f g )(x) = b. ( f g )(x) = c. ( fg)(x) = d. ( f/g)(x) = e. ( f g )(x) = f. f (g(x)) = g. g (f(x)) = h. ( g f )(x) = i. ( f g )() = j. f(g( )) = k. g (f()) = l. ( g f)( ) = m. f (x) = n. g (x) = o. f(f (x)) = p. g(g (x)) = q. f (f(x)) = r. f(f (x)) = 0. Show that f and g are inverses using function composition: f (x) = x 9 and g (x) = x 9. Find the value of y when x =. and y = 6x. Write the equation log8 = 5 in exponential form and explain how.. Write the equation ( ) x = 5 in logarithmic form and explain how.. Find the exact answers to the following (use properties) a. log ( 6 ) (hint: rewrite or use change of base) b. log 9 (hint: use the change of base formula) c. log55 0x d. log 5. Use the rules of logarithms to expand the following completely: x(x) a. log x b. og c. log x y 5 l b 9 z 8 x(x 5 ) (x) x 6. Use the rules of logarithms to condense the following completely: a. log(x ) log(x ) log log(x ) log5 b. l og5 log(6x 5) [log(x ) log(x )] 7. Find all real solutions to the following equation using properties of logarithms: a. l og(x ) l og = l og(5x ) b. log x log (x ) = c. ln(x ) l n(x ) = ln(x ) l n(x 7) 8. Find the vertical asymptotes, horizontal/slant asymptotes, holes, and x intercepts of the following, if any, after factoring: x a. f (x) = b. f (x) (x )(x 5) x 6x8 = x 6 x x 7x c. f (x) = x x (omit x intercepts part)
x 9. Divide f (x) = x a. Using synthetic division b. Using long division answer: x 8 x 7 x 0. Solve the following systems using the specified methods: a. Substitution: x y = 8 6x 5 y = b. Elimination: x y = x y = 0 c. Matrices: (record your steps) 5x z = 0 x y 5 z = 9 x y z = 0 d. Graphing: x y = x y = 6. Review how to add subtract and multiply matrices. Separate the following into partial fractions (factor first!): a. 5x x x answer: x x b. x x(x ) c. x 7x (x )(x x) answer: x x (x ) answer: 5 x x x x. Prove using mathematical induction: a. 6... n = n (n ) n(n) b. 5 7... ( n ) = Using arithmetic and geometric formulas for each of the following write their a) formula for the nth term in the sequence, a n =? b) calculate using the a formula c) formula for the nth term in the series, S n =? d) Calculate using the formula. 5, 8,,... a n = a 8 = S n = S = 5., 6,,... a n = a 8 = S n = S = 0 n=6 6. n Put the following into standard form then graph the equations on your own paper: a 8 s n 7. x y x 6 y = 5 8. y y 8 x = 0 9. x y 8 x 8 y = 0 0. x y 8 x 6 y = 0. Find each of the parts of the graphs below in inequality and interval notation, then describe how to answer these questions below:
Maximum(s): Maximum(s): Maximum(s): Maximum(s): minimum(s): minimum(s): minimum(s): minimum(s): To find the domain: To find the range: To find the intervals of increasing: To find the intervals of decreasing: To find the maximums: To find the minimums: ) Use Synthetic division to divide a) x x 5 x 5 by x x 7 x 6 x 5
b) 5x 6 x 8 by x 5x 0x 6 x ) Use long division to divide a) 6x x 5 x by x x x x b) x x 5 x 5 by x x 7 x 6 5 ) Use f = i and g = 5 i without calculators a) f g = 8 6i b) f g = i c) f * g = 7 6i d) f /g=? in standard form (a bi) /9 i/9 e) conjugate of f= i f) conjugate of g= 5i g) when are conjugates used? 5) Review adding, subtracting, multiplying, dividing, and factoring polynomials x ) Use Synthetic division to divide a) x x 5 x 5 by x x 7 x 6 x 5 b) 5x 6 x 8 by x 5x 0x 6 ) Use long division to divide a) 6x x 5 x by x x x x b) x x 5 x 5 by x x 7 x 6 5 ) Use f = i and g = 5 i without calculators a) f g = 8 6i b) f g = i c) f * g = 7 6i d) f /g=? in standard form (a bi) /9 i/9 e) conjugate of f= i f) conjugate of g= 5i g) when are conjugates used? 5) Review adding, subtracting, multiplying, dividing, and factoring polynomials x x
st Semester Pre Calculus Exam Review Key. cubic. a. x = /5 b. x= or x= c. x=, d. x=, e. Graph. a. y=/ b. y = c. y = d. (Hint: x= 0). a. (,9) b. ( 5, 9) c. ( 5, 9) 5 a. even power b. odd power c. odd end behavior d. even end behavior 6. a. 5/7(one solid point) b. b (solid point arrow to right) c. x=7, (hollow points shaded away) 7. x=8, 8. a. b. / 9 a. x x b. x x c. x x x d. x x e. x 9x f. x 9x g. 5 x h. 5 x i. j. 0 k. 7 l. 7 m. f (x) = ± x n. g (x) = x o.through r. = x 0. f(g(x))=g(f(x))=x. 7.89. 8 5 =. log/ 5 = x. a. b..8 c. 0x d. 6*8=8 5. a. l ogx l og(x ) log(x ) b. 9(logb x glogby 8logbz) c. l og l ogx 5log(x ) l og(x ) log(x ) 6. 5(x ) a. (x ) 5(6x 5) (x ) (x ) b. x 7. a. x = /9 b. x = 6, c. x = / 8. a. vertical: x=⅔, x=5 Holes: none Horizontal: y=0 x intercepts x= b. vertical: x= Holes: x= Horizontal: y= x intercepts: x= c. vertical: x=, Holes: none Slant: y = x 9. a=b= x 8 x 7 x 0 a. ( 5, ) b. (, ) c. rref d. (, ). Review how to add subtract and multiply matrices. a. x x b. x x c. 5 (x ) x x x x. a. True b. True. 5, 8,,... a n = 5 (n ) a 8 = 6 S n = n (5 a n ) S = 8 5., 6,,... a n = * n a 8 = 8 6. 655 S n = * n S = 5 Put the following into standard form then graph the equations on your own paper: (y ) 7. = 8. x = (y ) (x ) 8 9. ( x ) ( y ) = 8 (x ) 0. (y ) =
inequality: y>= Interval: (, ) inequality: y>= 6 Interval: ( 6, ) inequality: y<=75 Interval: (, 75) x>0 Interval: ( 0, ) inequality: x> Interval: (, ) inequality: x<, <x< Interval: (, ) (, ) x<0 Interval: (, 0) inequality: x< Interval: (, ) none inequality: x>, <x< Interval: (, ) (, ) Maximum(s):none Maximum(s):none Maximum(s):none Maximum(s): (,5),(,75) minimum(s): (0, ) minimum(s):(, 6) minimum(s):none minimum(s): (.75, 0) To find the domain: x values minmax To find the range: y values minmax To find the intervals of increasing: up (refers to x values) To find the intervals of decreasing: down (refer to x values) To find the maximums: (highest points, change from increasing to decreasing) To find the minimums: (lowest points, change from decreasing to increasing)