3. Solve the following. Round to the nearest thousandth.

Transcription

1 This review does NOT cover everything! Be sure to go over all notes, homework, and tests that were given throughout the semester. 1. Given g ( x) i, h( x) x 4x x, f ( x) x, evaluate the following: a) f hx b) h gx. If f x x and g x x, determine if f and g are inverse functions.. Solve the following. Round to the nearest thousandth. a) 1 x4 x 1 b) log m (16) 5log m () log m ( x) c) e t

2 4. For the following rational function, determine the coordinates of any removable discontinuity, the equations of any vertical, horizontal, or slant asymptotes, the coordinates of any x or y intercepts and then sketch a graph. Please label all important information. f x x x 1x 18 5x 6 5. Graph the following function: f x ( x )( x 6x 5) x 4x4

3 Given the parabola y x x. First, put into vertex form by completing 1 the square. Then state the vertex, focus, equation of the directrix, and width. Then sketch a graph. 8. Solve the following. Show all work. x b) 56 a) 5 x 5 10x 64 4 x

4 9. Given the equations of two circles shown below, put the equations into standard form, then determine the center and radius of the circle. a) x y 6x 8y 1 0 b) 4x 4y 40x 4 y Simplify: y 7 y a) 4 y y 81y 19 y b) x 1x x 75x x 108 x Solve and check in the original equation. a) x x 18 b) 7 6x x

5 1 x Solve. Express your answer in exact form x 5 1. Simplify. Express your answer in exact form. 5 x Simplify. (Hint: Start with in the parentheses) Solve for x. 1 x x 9 x 6 x

6 16. Find the third degree polynomial equation with the roots 1 1 i and 17. Solve for the variable indicated in the parenthesis. a) S L rl ( L ) b) c d a ( c ) 18. Write the equation of each function in the graph below. a. Equation: Domain: Range: b. Equation: Domain: Range: c. Equation: Domain: Range: d. Equation: Domain: Range:

7 1 19. Solve the following: log 4( x ) log 4 4 log Solve the following: ln(x 1) 6 1. A rocket is launched into the air and at the following times: 15, 45, 70, 85 seconds, reaches the corresponding heights: 000, 4400, 40700, 9000 feet. a) Determine the quadratic function which best describes the path of this rocket. b) What is the rocket s height at 10 seconds? c) At what two times will this rocket reach a height of feet? d) When will the rocket return to the ground? e) At what time will the rocket be at its maximum height? What is the maximum height?. Write the standard and general equation of a circle that has a diameter with endpoints of (4, 8) and (-, 0). Given (-9, -4), (-, ) and (, -4), find the standard equation of the circle that passes through these three points. Then state the center and radius of the circle.

8 For numbers 4 and 5, find all vertical asymptotes, horizontal asymptotes, x intercepts, y intercepts and any points of discontinuity that exist. Then, draw a sketch of the function. x 4x 4. f ( x) x x 6 5. f ( x) x x Find all vertical asymptotes, horizontal asymptotes, x intercepts, y intercepts and any points of discontinuity that exist. DO NOT DRAW A SKETCH! f ( x) x x x x

9 7. Simplify. a) 7 x x x 40x 45x 8 7 b) 9 7x y 8x z 6 8. Find the equation of a parabola with a horizontal directrix that passes through the points (5, ), (-, 51) and (0, 18). Then rewrite the equation into standard form. 9. Write the general and standard equation of a parabola that has a vertex at (-1, ), passes through (, 6) and has a vertical directrix.

10 0. Find the coordinates of the focus and the vertex and the equations of the directrix and axis of symmetry for the parabola with the equation below. Then determine the width of the parabola through the focus and graph the parabola. a. ( x ) 8( y 4) b. ( y 1) 0( x 9) 1. Factor Completely x 15 x

11 . Solve for x using logarithms. x x 5 4. Solve for x. log 5x log 9 6 log 9 9 x 4. Graph the following piecewise function, and answer the questions that follow. f( x) x x, x, x Function? Yes or No f f f 1 0 f ( x ), x Domain : Range :

12 5. Graph the following: x y y 6 8 Direction of opening: Vertex: p = Focus: Directrix: Axis of Symmetry: 6. Rewrite the equation in standard form, then graph. y x 490x Center: Major axis // to: Length of major axis = Endpoints of major axis = Length of minor axis = Endpoints of minor axis = Foci =

13 7. Given the following equation, graph. Center: Major axis // to: Length of major axis = Endpoints of major axis = Length of minor axis = Endpoints of minor axis = Foci = ( x) ( y9) Rewrite the equation in standard form, then graph. x y 1y 19 18x Center: vertices: Foci = Asymptotes =

14 9. Given the following equation, graph. ( y1) ( x1) Center: vertices: Foci = Asymptotes = 40. Evaluate the following limits. a. lim x7 x x 7 b. x lim x0 x x c. lim x x x 8 x d. lim x 4 x 16 e. x 9 lim x x

15 41. Use the graph below to evaluate the following limits. a. lim f x x0 d. lim f x x b. lim f x x0 e. lim f x x c. lim f x x0 f. lim f x x g. g h. lim f x x4 4. In the geometric series, the first term is 100 and the third term is 1. i. Find the common ratio; ii. Write in sigma notation; iii. Find the sum of the infinite series if possible.

16 4. In an arithmetic sequence, the eighth term is -5 and the hundredth term is 4. Find the first term, and the 00 th term. 44. Given the sequence: 1 1 1,,,...,, find the number of terms in the sequence Which term is 1 in the following sequence: 768, 819, 048,.? In the series: , Sn 7. Find the number of terms.