Math 119 Pretest Review Answers

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1 Math 9 Pretest Review Answers Linear Equations Find the slope of the line passing through the given points:. (, -5); (0, ) -7/. (, -4); (, 0) Undefined. (-, -); (, 4) 4. Find the equation of the line with slope that passes through (-,-). Write your answer in slopeintercept form. y + 5. Find the general form of the equation of the line that passes through (-,6) and is parallel to the graph of + y y Find the general form of the equation of the line that passes through (4,-) and is parallel to the graph of y 0. y 6 0 Quadratic Equations Rewrite the following equations in general quadratic form: Complete the square for the following functions: ( ) Factoring Factor: ( + ) ( ). 4 y 6 y 8y 4y ( 4 7) ( + )( 7) 5. 5y + 5y + a + a (Group in pairs) ( + )(5y + a) 6. ( + ) 7. a y 64 ( + )( ) ( a 4)( a + 4a + 6) ( y + 8)(y 8) Finding Zeros/Roots 9. Use Descarte s Rule of Signs to determine the possible number of positive and negative zeros: + 0 positive, negative Page

2 4 0. Given + 6 5, determine the possible number of negative zeros. Eactly One. List the possible rational zeros of the function: ±, ±, ±, ±. List the possible rational zeros of the function: ±, ±, ±, ± 6, ±, ±, ±, ± 4 4. Use the fact that i is a zero of f to find the remaining zeros: ,, ± i 4. Find all of the zeros of the 4 function: ± i, ± 4i 5. Find all the real zeros of the polynomial function: g ( t) t + t 6t 48-4, -, 4 6. Use the Intermediate Value Theorem to estimate the real zero in the interval [, ]: Between. and. 7. Use the Intermediate Value Theorem to estimate the real zero in the interval [, 4]: Between.0 and. 8. Use the Intermediate Value Theorem to estimate the real zero in the interval [0, ]: Between 0.9 and.0 Long and Synthetic Division 9. Find all of the real roots: ,, 0. Find all of the real zeros of the function: 7 + 7,,. Write as a product of linear factors: ( 4)( + ) +. Simplify the rational epression: Simplify the rational function: Simplify: + + ( )( ) 4 Absolute Value Solve the following equations for : {,5} {-5, 5} No Solution Page

3 Simplify the following epressions involving absolute values: Correction: ( ) Pythagorean Theorem For questions 9-0, consider a right triangle with sides of length a, b, and c: 4. If the lengths of sides a and b are si and eight inches respectively, how long is side c? 0 in. 4. What is the length of side b in terms of the sides a and c? c Distance Formula Find the distance between the given points: 4. (, 5); (-,9) 5 a 44. (-, ); (5,) (0, -); (, 0) 58 Simplifying Epressions Simplify: (Assume all variables represent positive 7 y real numbers) ( y ) 7 y y 49. y ( ) ( 4) y ( + ) ( ) ( ) Page

4 Rationalize the denominator and simplify: Solving Equations Solve for or y: 55. ( + 5) 4 {-, } ± ± No solution y + y y No Solution y + y y y + y y {0, 4} {-, } Inequalities Solve each inequality: Know both forms of the answer! 67. > < or (,) 68. < 4 + >, or ( ) or [, ) 70. > 0 6 <, < < or (, ) (,) Page 4

5 < or [,) 7. 0 < 0, or [,0 ) [, ) 7. + > + <, > or (, ) (, ) , 0 4 or (, ] [ 0,4] < < <, > or (,) (, ) Solve each absolute value inequality: Know both forms of the answer! 76. < < < 5 or (,5 ) 5 5, or (, 5] [, ) Logarithmic, Eponential and Logistic Functions 79. Write as the logarithm of a single quantity: [ ln( + ) + ln( ) ] + ln ln( )( + ) ( ) 80. Write as the logarithm of a single quantity: ln + ln + ln y ( ) 9 + ln y Simplify: 8. e ln( + ) 8. ln 5e ( +) + ln 5 8. log log log The spread of a flu virus through a certain 000 population is modeled by y 0. 7t + 990e where y is the total number infected after t days. In how many days will 80 people be infected with the virus? days 86. The amount remaining of a decaying sample is t h described by the function m( t) m0 where m 0 is the initial amount, t is the number of hours, and h is the sample s half-life. Given an isotope of sodium with a half-life of 5 hours, find the amount remaining from a g. sample after 0 hours g Page 5

6 87. The number of bacteria in a culture is given by 0.5t the formula n( t) 00e What is the half-life for this type of bacteria? t Sketch the graph of the following +, function:, > Conic Sections 88. Find the standard equation of the parabola with verte at (0,0) and directri 7 y Find the standard equation of the ellipse with center at (0,0) a focus at ( 5,0) and minor ais of length 4. + y Find the foci of the hyperbola: y Graph the following function: ( 0, ± ) Graphing and Translations 9. Determine the correct function for the given graph. 94. Use the graph of function y f (). 4 y to find a formula for the (a) (b) f 4 ( ) (c) + (d) f ( ) 4 y y f () (e)none of these (d) y ( ) 4 + Page 6

7 95. Given the graph of y, graph y +. Asymptotes 00. Graph the following function: + y 0. Graph the following rational function: What sequence of transformations will yield the graph of g ( ) ( + ) + 0 from the graph of ( ) f? Horizontal shift unit to the left, vertical shift 0 units up 0. Match the graph with the correct function. Domain Determine the domain of the function: (,4) (4, ) (,6] f ( ) if > if <- (, ] (, ) 5 (a) (b) + ( ) + 5 (c) (d) + (e) None of these (c) Page 7

8 End Behavior Describe the end behavior of the following functions: The end behavior of f () is that it follows the line y. f () as and f () as g ( ) The end behavior of g() is that it follows the parabola y ; g () as ± h ( ) The end behavior of h() is that it approaches the line y ; h ( ) as ± Vertical and Horizontal Line Tests 06. Use the vertical line test to determine in which case y is a function of. 07. Which of the following graphs represent y as a function of? (a) One-to-One Functions Determine whether each function is one-to-one. If it is, find its inverse yes, it is; f 7 ( ) 09. y + Not one-to-one Inverse functions Find the inverse of the following functions. State the domain and range of f ( ). 0. +, for 0 (d) f ( ) ; Domain: (,] ; Range: [ 0, ) Page 8

9 ., for 0 Range: [ 0, ) ; Domain: [ 0, ) ; , for 0 5 f ( ) 5 Domain: [ 0,5] ; Range: [ 0,5] Composition of Functions. Evaluate ( f o g)() where 4 and g ( ) Evaluate ( f o g)() where + 7 and g ( ) Evaluate ( f o g)() where + 7 and g ( ) Partial Fraction Decomposition Find the partial fraction decomposition for the following epressions: ; Sequences and Series 9. Write the first five terms of the geometric sequence with a and r ,,, 9, 7 0. Find the eighth term of the arithmetic sequence with a 5 and d 8. (Assume that n begins with.) 6. Determine whether the following is a geometric series, if so, find r: -, 4, -8, 6, - Yes, r - Probability. The flags of seven different countries are to be displayed in a row. In how many different orders can they be flown? Nine candidates are participating in an election. In how many different orders can they finish? 6, There are eight football teams in the Mountain West Conference. In how many different orders can these eight teams finish in the final rankings? Si friends are driving to a basketball game. In how many different orders can they arrive? A bo holds white, 5 red, and 6 black marbles. If marbles are picked at random, without replacement, what is the probability that they will both be black? 5 5 Page 9

10 Comple Numbers Simplify the following comple numbers: 7. i + i + + i 8. i i + 4i + i 4 9. i + 4i i i + 4 7i Find the conjugate of the following comple numbers: i. 6. i i i Page 0