1 x. II. CHAPTER 2: (A) Graphing Rational Functions: Show Asymptotes using dotted lines, Intercepts, Holes(Coordinates, if any.)

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1 FINAL REVIEW-014: Before using this review guide be sure to study your test and quizzes from this year. The final will contain big ideas from the first half of the year (chapters 1-) but it will be focused on the last two marking periods. I. CHAPTER 1: Functions: Determining if a relation is a function, finding domain and range, zeros, intercepts and testing for symmetry. Determining if a function is even or odd. Continuity, end behavior (with notation) and limits. Parent functions and transformations. Inverse Functions. Determine if the function is even/odd/neither: 1. Find. f ( x) x 9 4 y x x f 1 ( x). 4. x. gx ( ) 1 x x x 5 f( x) 5. f( x) x x 9 II. CHAPTER : (A) Graphing Rational Functions: Show Asymptotes using dotted lines, Intercepts, Holes(Coordinates, if any.) 1. Find all asymptotes of the graph of each rational function: a. f(x) = x b. f(x) = x x +1 x 1. Find all asymptotes and holes: f(x) = x +x x x 6. Sketch the graph of the rational functions: a. f(x) = x 1 d. f(x) = x x x+1 x b. f(x) = x x x c. f(x) = x 9 x x (B) Finding zeros of Polynomials: 1. Write the polynomial f(x) = x 4 x 0: (a) as the product of factors that are irreducible over the rationals (b) as the product of linear factors and quadratic factors that are irreducible over the reals, and (c) in completely factored form.. Find a polynomial function with real coefficients that has the given zeros (there are many correct answers):,, 4 i. Use the given zero to find all zeros of the function: g(x) = x 4x + 8x + 8 Given zero: 1 i. II. CHAPTER : EXPONENTIAL/LOGARITHM FUNCTIONS: Solving Exponential and Log Equations, and Graphing same.condensing/expanding/evaluating log expressions (A) Solve for x: 1. 1 x. 64 x 1 x x 8 x 9

2 (B) Note: All solving will be exact form, as no calculators will be used. 1. Match the function with its graph for the following: i. f(x) = 4 x ii. f(x) = 4 x iii. f(x) = 4 x iv. f(x) = 4 x + 1. Graph the function by hand, identify any asymptotes and intercepts and determine whether the function is increasing or decreasing: a) g(x) = 6 x b) f(x) = 0. x. Find the domain, vertical asymptote, and x-intercept of the logarithmic function and sketch its graph by hand: a) f(x) = log (x 1) Use the properties of logarithms to rewrite and simplify the logarithmic expression: a) ln(5e ) b) log Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms (assume all variables positive): a) log 5 y x+ b) ln c) ln xy5 x xy z 6. Condense the expression to the logarithm of a single quantity: a) 1 ln(x 1) ln(x + 1) b) ln + 1 ln(4 x ) ln x c) [ln x ln(x + 1)] + ln 5 7. Solve the equation for x without using a calculator: a) log (x 1) = b) ln(x + 1) = 4 8. Solve the exponential equation algebraically: a) e 5x = 1 b) 14e x+ = 560 c) 4(5 x ) = 68 d) e x 7e x + 10 = 0 9. Solve the logarithmic equation algebraically: a) log 10 (x 1) = log 10 (x ) log 10 (x + 1) b) log 10 (x + ) log 10 x = log 10 (x + 5) 10. Find the exact value of the logarithm without using a calculator: a) log ( 4) b) log 5 75 log 5 c) ln e ln e 7 d) ln e 4 e) ln 1 e

3 III. CHAPTER 4: TRIG FUNCTIONS: This supplements the Review Hand-out on Ch 4 Note: All exact answers, as no calculators will be used. 1. Given a point on the terminal side of an angle in standard position, determine the exact values of the 6 trigonometric functions: a) (5, 1) b) ( 4, 10). Find the values of the 6 trigonometric functions of θ. a) csc θ = 4 given cot θ < 0 b) sin θ = 0 given π θ π c) tan θ is undefined given π θ π. Evaluate the trigonometric function of the quadrantal angle. a) sec π b) cot π c) sec 0 d) cot π 4. Evaluate the sine, cosine, and tangent of the angle (without a calculator). a) 750 b) 40 c) 11π d) 17π Find two solutions of the equation. Give you answer in radians (0 θ < π) a) sin θ = 1 b) csc θ = c) cot θ = 1 d) sec θ = 6. Find the exact value: a) tan (arccos ) b) cos [arcsin ( )] 5 7. Write each of the following as an algebraic expression in terms of x: a) sin(arccos x) 0 x 1 b) cot(arccos x) 0 x 1 c) csc (arctan x 7 ) 8. Find the exact value of the expression without using a calculator. a) arcsin 1 f) arccos ( 1 ) b) arcsin 0 c) arctan d) arctan( 1) e) cos 1 ( ) 9. Use the law of sines to solve the triangle. If solutions exist, find both. a) A = 75, a =.5, b = 16.5 b) B = 115, a = 9, b = 14.5 c) A = 15, a = 5, b = A tree stands on a hillside of slope 8 from the horizontal. From a point 75 feet down the hill, the angle of elevation to the top of the tree is 45. Find the height of the tree. (see diagram) 11. Use law of cosines to solve the triangle: a) a = 9, b = 1, c = 0 b) B = 150, a = 10, c = 0 1. Two planes leave Washington, D.C. s Dulles International Airport at approximately the same tiem. One is flying at 45 miles per hour at a bearing of 55, and the other is flying at 50 miles per hour at a bearing of 67. Determine the distance between the planes after they have flown for hours.

4 1. Find the area of the triangle given: a) a = 4, b = 5, c = 7 b) A = 7, b = 5, c = 8 IV. CHAPTER 5: VERIFYING TRIG IDS AND SOLVING TRIG EQNS. Use Hand-out as is. V. CHAPTER 6: SYSTEMS OF EQUATIONS AND MATRICES 1. Use matrices to solve the system of linear equations: a) { x + y z = 1 x 5z = b) { x y + z = 6 x y = 7 4x y z = 14 x + y z = 11. Use Gaussian or Gauss-Jordan Elimination to solve the system: x + y + z = x + 1y 9z = 1 a) { 6x + 6y + 1z = 1 b) { x + 15y 1z = 0 1x + 9y z = 1. Find AB if possible: a) A = [ 5 4], B = [ ] b) A = [ ], B = [ 5 ] Find the inverse of the matrix if it exists: a) [ ] b) [ ] c) [ 6 ] Find the determinant of the matrix: a) [ ] b) [ 6 0 ] c) [ ] Use Cramer s Rule to solve the system: x + y 5z = 11 x + y = 5 a) { b) { 4x y + z = x + y = 1 x 4y + 6z = 15 CHAPTER 7 For each conic re-write into standard form, sketch the graph and then provide the important information. Circle: center and radius Parabola: vertex, focus, directrix, axis of symmetry Ellipse: center, vertices, co-vertices, foci, and eccentricity Hyperbola: center, vertices, foci, and equations of asymptotes 1. y x 1y x 4y 4x 1y x 16y 6x 80y x 16y x 40 y 5x 9y 150x 6y 6 0

5 7. x 18x y 0 8. x y x y Use the information provided to write the standard form equation of each circle. 9. The endpoints of the diameter are (1, 5) and (-,-5). 10. The center is at (9, 5) and passes through the point (16,-). 11. The center lies on the y-axis and is tangent to the x-axis and the line y 10. Use the information provided to write the standard form equation of each parabola. 1. The vertex is (-7,-) and the focus is 7, The focus is at,0 4 and the directrix is 19 x 4 Use the information provided to write the standard form equation of each ellipse Use the information provided to write the standard form equation of each hyperbola Part II: Sketch the curve given by each pair of parametric equations over the given interval. 1. x = t + 1 and y = t 6; 5 t 5 Write each pair of parametric equations in rectangular form.. x = t +, y = t 4. x = t + 5, y = t 4. x = sin θ, y = cos θ

6 5. Write two sets of parametric equations for the given rectangular equation: y = x BASEBALL Micah hit a baseball at an initial velocity of 10 feet per second from a height of feet at an angle of 4. a. How far will the ball travel horizontally before it hits the ground? b. What is the maximum height the ball will reach? c. If the fence is 8 feet tall and 400 feet from home plate, will the ball clear the fence to be a home run? Explain. CHAPTER 9: POLAR COORDINATES AND COMPLEX NUMBERS Part I: 1. Find the polar coordinates that do not describe the point in the given graph. A ( 4, 10 ) C (4, 150 ) B (4, 0 ) D (4, 0 ). Find the equation represented in the given graph. F θ = π H θ = G r = π J r = π. AIRPLANES Two airplanes at the same altitude have polar coordinates (.5, π 6 ) and ( 1.9, π ), where r is in miles. Find the distance between them. A.49 miles B.14 miles C.91 miles D 1.65 miles 4. Find the equation whose graph is given. F r = 4 cos θ H r = 4 cos θ G r = + cos θ J r = 16 cos

7 5. Identify the graph of the polar equation r = 4 sin θ. A B C D 6. Find polar coordinates for the point with rectangular coordinates (, ) if 0 θ π and r 0. F (4, π ) G (4, π ) H (4, 5π 6 ) J (, π ) 7. ROBOT A robotic vacuum is positioned so a point on it has polar coordinates (4, 5π ). Find rectangular 4 coordinates for the point. A (, ) B (, ) C (, ) D (, ) 8. Write the rectangular equation x + y x = 0 in polar form. F r = sin θ G r r sin θ = 0 H r = cos θ J r = cos θ 9. Write the polar equation r r sin θ = 0 in rectangular form. A x + y = 0 C x + y y = 0 B x + y x = 0 D x = y 10. Express 5 5i in polar form. A 10 (cos 11π + i sin 11π 11π ) C 10 (cos B 5 (cos 11π 6 i sin 11π 6 ) + i sin 11π 6 ) D 10 (cos 5π + i sin 5π ) 11. Express 4 (cos π + i sin π ) in rectangular form. 4 4 F + i G i H i J + i For Questions 17 and 18, let z 1 = 8 (cos π + i sin π ) and z = 0.5 (cos π + i sin π ). 1. Write the rectangular form of z 1 z. A 4i B 4 C 4 + 4i D 4 1. Write the rectangular form of z 1 z. F i G i H i J 8 8 i 14. Which of the following is not a third root of 1 i to the nearest hundredth? F i G i H i J i

8 Part II: 1. Plot the complex number and find its absolute value: a) 7i b) 5 + i. Write the complex number in trigonometric form: i. Perform the operation and leave the result in trigonometric form: a) [ 5 (cos π + i sin π )] [4 (cos π + i sin π 0(cos 0 +i sin 0 ) )] b) 4 4 5(cos80 +i sin 80 ) 4. Find the indicated power of the complex number: a) [5 (cos π 1 + i sin π 1 )]4 b) ( + i) 6 5. (a)find the indicated roots of the complex number (b) represent the number graphically (c) write the number in rectangular form: i) 6 th roots of 79i ii) cube roots of 8 And (Thicker Final Review Hand-out Chapter 6): Just 9-11;17-

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