STUDY GUIDE ANSWER KEY 1) (LT 4A) Graph and indicate the Vertical Asymptote, Horizontal Asymptote, Domain, -intercepts, and y- intercepts of this rational function. 3 2 + 4 Vertical Asymptote: Set the denominator equal to zero and solve for. This is where is undefined because a denominator cannot equal zero. 2 + 4 = 0 2 = 4 = 2 Horizontal Asymptote: Eponents are the same so divide coefficients (EATS DC). y = 1 2 Domain will be determined by the vertical asymptote. (, 2) ( 2, ) The y-intercepts can be found by substituting zero for. 0 3 2(0) + 4 = 3 4 The -intercepts can be found by setting the numerator equal to zero. 3 = 0 = 3
2) (LT 4A) Graph and indicate the Vertical Asymptote, Horizontal Asymptote, Domain, -intercepts, and y- intercepts of this rational function. 2 2 8 2 + 3 10 Vertical Asymptote: Set the denominator equal to zero and solve for. This is where is undefined because a denominator cannot equal zero. 2 + 3 10 = 0 ( + 5)( 2) = 0 + 5 = 0; 2 = 0 = 5, 2 Horizontal Asymptote: Eponents are the same so divide coefficients (EATS DC). y = 1 1 = 1 Domain will be determined by the vertical asymptote. (, 5) ( 5,2) (2, ) The y-intercepts can be found by substituting zero for. 0 2 2(0) 8 0 2 3(0) 10 = 8 10 = 8 10 = 4 5 = 0.8 The -intercepts can be found by setting the numerator equal to zero. 2 2 8 = 0 ( 4)( + 2) = 0 4 = 0; + 2 = 0 = 4; = 2
3) (LT 4B) Graph and indicate the Vertical Asymptote, Horizontal Asymptote, Domain, -intercepts, and y- intercepts of this rational function. 2 6 + 5 3 Vertical Asymptote: Set the denominator equal to zero and solve for. This is where is undefined because a denominator cannot equal zero. 3 = 0 = 3 Horizontal Asymptote: Bigger on Top, None (BOTN) No Horizontal Asymptote -intercept: The -intercepts can be found by setting the numerator equal to zero. 2 6 + 5 = 0 ( 5)( 1) = 0 5 = 0; 1 = 0 = 5; = 1 Domain will be determined by the vertical asymptote. (, 3) (3, ) y-intercept: The y-intercepts can be found by substituting zero for. 0 2 6(0) + 5 = 5 0 3 3 = 5 3 Oblique Asymptote: To find the oblique asymptote, we need to divide the rational function using synthetic division. The quotient we get is our linear function. 3 1-6 5 3-9 1-3 -4 y = 3
4) (LT 4C) Factor 3 64 Use the difference of cubes formula: a 3 b 3 = (a b)(a 2 + ab + b 2 ) 3 64 = ( 4)( 2 + 4 + 16) 5) (LT 4C) Simplify 3 + 27 2 9 We can use the sum of cubes formula for the numerator: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) And the difference of squares formula for the denominator: a 2 b 2 = (a b)(a + b) Therefore we would get: ( + 3)( 2 3 + 9) ( + 3)( 3) ( 2 3 + 9) ( 3) 6) (LT4C) Simplify 2 10 + 25 5 + 3 + 3 We can factor the numerator and flip the second fraction and change this problem into a multiplication problem. ( 5)( 5) 7) (LT 5A) If $3,000 is invested in an account paying 8% compounded daily, how much will be in the account at the end of 10 years? Compute the answer to the nearest cent. We use the compound interest formula: A = P (1 + r n )nt. P = 3000, r = 0.08, n = 365, t = 10 A = 3000 (1 + 0.08 365 ) 365(10) A = $6676.04 Note that n is 365 because it is compounded daily. The rate is 0.08 because we need to convert 8% into decimal form. + 3 ( 5)( 5) + 3 5 + 3 5 + 3 5 8) (LT 5A) Meico has a population of around 60 million people, and it is estimated that the population will double in 23 years. If population growth continues at the same rate, what will be the population 12 years from now? We use the population formula: P = P 0 (2) t d P 0 = 60 million, d = 23 years, t = 12 P = 60 (2 12 23) = 86.14 million people 9) (LT 5B) Rewrite in equivalent eponential form. log 4 16 = 2 10) (LT 5B) Rewrite in equivalent logarithmic form. 125 = 5 3 4 2 = 16 11) (LT 5C) Write in terms of simpler logarithmic form log b 3 y 4 Property: log b M N = log b M + log b N log b 3 + log b y 4 Property: log b M p = P log b M 3log b + 4log b y log 5 125 = 3 12) (LT 5C) Write in terms of simpler logarithmic forms log b yz q Property: log b M N = log b M + log b N Property: log b ( M N ) = log bm log b N log b + log b y + log b z log b q
13) (LT 5C) Write in terms of simpler logarithmic forms 2log b + log b y log b z Property: log b M N = log b M + log b N Property: log b ( M ) = log N bm log b N Property: log b M p = P log b M 14) (LT 5C) Solve for without using a calculator. log 2 ( + 4) = 2log 2 4 Property: log b M p = P log b M log 2 ( + 4) = log 2 4 2 Logs are equivalent so we can cancel them. + 4 = 16 = 12 log b 2 y z 15) (LT 5C) Solve for without using a calculator. log 3 = log + log( + 2) log 3 = log ( + 2) Logs are equivalent so we can cancel them. 3 = ( + 2) 3 = 2 + 2 0 = 2 + 2 3 Factor 0 = ( + 3)( 1) = 3, 1 17) (LT 6A) Find measure of the angle indicated 16) (LT 6A) Find the value for. First, label which side is the Opposite, Adjacent, and Hypotenuse. The opposite side is because it is opposite of the angle and the adjacent side is net to the angle. The hypotenuse is 10.3 because it is the side across from the right angle. Using SOH CAH TOA, we can see that we have the hypotenuse and opposite side letting us use sine. Sine is the relationship between the opposite and the hypotenuse side. sin 37 = 10.3 10.3 sin 37 = 6.20 = First, label which side is the Opposite, Adjacent, and Hypotenuse. The opposite side is 7.7 because it is opposite of the angle and the adjacent side is 14 because it is net to the angle. The hypotenuse is the side across from the right angle. Using SOH CAH TOA, we can see that we have the adjacent and opposite side letting us use tangent. Tangent is the relationship between the opposite and the adjacent side. tan θ = 7.7 14 tan 1 (tan θ) = tan 1 ( 7.7 14 ) θ = 28.81 18) (LT 6A) You are standing 15 feet away from a tree, and you measure the angle of elevation to be 38. How tall is the tree? First, draw the triangle. The adjacent side is 15 feet, the angle is 38, and the opposite side is. Using SOH CAH TOA, we can see that we have the adjacent and opposite side letting us use tangent. Tangent is the relationship between the opposite and the adjacent side. tan 38 = 15 15 tan 38 = 11.72 feet = 15 38
19) (LT 6A) A dog, who is 10 meters from the base of a tree, spots a squirrel in the tree at an angle of elevation of 40. What is the direct-line (diagonal) distance between the dog and the squirrel? First, draw the triangle. The adjacent side is 10 feet, the angle is 40, and the hypotenuse side is. Using SOH CAH TOA, we can see that we have the adjacent and hypotenuse letting us use cosine. Cosine is the relationship between the adjacent side and the hypotenuse. cos 40 = 10 cos 40 = 10 = 10 = 13.05 meters cos 40 10 40 20) (LT 6B) Find AB 21) (LT 6B) Find the measure of the angle of C Using the Law of Sines, sin A sin B sin C = = a b c We can find the missing side or angle of a non-right triangle. sin 44 sin 53 = 7 c Solving for c: c sin 44 = 7 sin 53 7 sin 53 c = sin 44 c = 8.05 22) Fill in the Unit Circle Using the Law of Sines, sin A sin B sin C = = a b c We can find the missing side or angle of a non-right triangle. sin 88 26 Solving for angle C: 21(sin 88) = sin C 21 = sin C 26 sin 1 21(sin 88) ( ) = C 26 53.82 = C