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Download PRACTICE TEST #4 – FULL ANALYSIS Topics to Know Explanation
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Topics to Know PRACTICE TEST #4 β FULL ANALYSIS Explanation 1. How to factor 2. Inverse Functions Switching logs to exponents (ππππ π = π β π π = π) Multiplying binomials (FOIL) Write it out as two binomials first, and FOIL Converting imaginary numbers carefully. Remember that no βpowersβ of π can be in your final answer. π 2 = β1 3. 4. 5. 6. 7. 8. How to work with complex fractions GCF always goes first, if possible. Then, case I or case II trinomial factoring To take the inverse of a function, (1) switch x and y, then (2) re-solve for y Two steps (1) find the LCD of all little fractions, (2) multiply each piece by the FULL LCD and cancel what you can. The point is to get all of the βlittleβ fractions to cancel out. Solving inequalities and testing Set equal to 9 to find your critical points. Think whatβs inside the absolute value brackets can be equal to 9 (because the absolute value of 9 is 9), or it could be equal to -9 (because the absolute value of -9 is 9) Once you have the critical points, test any number youβd like in each of the three sections back into the original inequality to see what sections are true, and which are false Calculator Skills! Enter the data into List1 Run 1-Var Stats Sample standard deviation is πΊπ₯ Domain Restrictions Domain is all x-values. You can plug in any Denominators β 0 real number you can think of for x, unless it Radicals β₯ 0 makes this denominator equal zeroβ¦ thatβs a no-no Composition of functions Plug 5 into the f function first. Then plug Work from the inside out that answer into the g function Work 2 5π₯(π₯ β 4π₯ + 3) = 0 5π₯(π₯ β 3)(π₯ β 1) = 0 π¦ = πππ3 π₯ π₯ = πππ3 π¦ (switch x and y) 3π₯ = π¦ (convert log into exponent to re-solve for y) (3 β 5π)(3 β 5π) 9 β 15π β 15π + 25π 2 9 β 30π + 25π 2 9 β 30π β 25 β16 β 30π 1 +1 π₯ 1 β1 π₯ LCD: x 1 (π₯) + (π₯)1 π₯ 1 (π₯) β (π₯)1 π₯ 1+π₯ 1βπ₯ 2π₯ + 3 = 9 2π₯ + 3 = β9 π₯=3 π₯ = β6 Test: π₯ = β8 π₯=0 |2(β8) + 3| < 9 |2(0) + 3| < 9 13 < 9 3<9 False TRUE 2π₯ + 10 β 0 π₯ β β5 π(5) = 4(5) β 5 = 15 π(15) = β2(15) + 7 = βππ π₯=4 |2(4) + 3| < 9 11 < 9 False Topics to Know PRACTICE TEST #4 β FULL ANALYSIS Explanation 9. Simplifying Radicals Break each individual radical down. Terms can be combined when the variables and the radicals match 10. Function: No x-values repeat One-to-one: No y-values repeat 11. (Reference sheet formula) Look for an answer choice that does not have any repeating yβs (one-to-one) and does not have any repeating xβs (function) Draw & Label. Ensure that the angle used is in between the two sides used. 12. 13. 14. Function: No x-values repeat Calculator skills! Multiplying Binomials Pythagorean Trig Identities 15. Shifts 16. Sum & Product formulas 17. (Reference sheet formula) 18. 19. Work 5βπ₯ 2 β5π₯ 4π₯β4β5π₯ β 4π₯ β 2β5π₯ β 5 β π₯β5π₯ 8π₯β5π₯ β 5π₯β5π₯ 3π₯β5π₯ 1 π΄πππ = 2 (23)(14)(sin 71) π΄πππ = 152.2 Vertical line test Multiply them out carefully Then, using Pythagorean identities (you know itβs pythag because thereβs a trig function squared), turn it into one trig function The number added/subtracted attached to the x is your left & right movement (itβs backwards) The number added/subtracted outside of the x is your up & down movement π π Sum = β π Product = π (1 β cos π₯)(1 + cos π₯) 1 β cos2 π₯ Pythag Identity: sin2 π₯ + cos 2 π₯ = 1 sin2 π₯ = 1 β cos 2 π₯ sin2 π₯ β(β4) 1 9 = 41 Sum = Use a quick right triangle & pythag theorem πππ to find the value of sin (π»π¦π) cos π₯ Switch from log to exponent Variable stuck in a logβ¦ switch to exponent Arc Length formula Remember the angle must be in radians (it usually is) 51 = π₯ + 4 5=π₯+4 π₯=1 π π=π =4 8 1 Product = = 8 40 sin π₯ = 41 sin 2π₯ = 2(sin π₯)(cos π₯) 40 9 720 sin 2π₯ = 2 (41) (41) = 1681 π 2 = 10 π = 20 Topics to Know PRACTICE TEST #4 β FULL ANALYSIS Explanation 20. Review packet, Trig page, bottom left cornerβ¦ SO IMPORTANT!!! Isolate (this one is simple solving)β¦ ref angleβ¦ quadrantsβ¦ put it all together 21. Axis of symmetry formula π π₯ = β 2π (the non-radical part of the quadratic formula!) 22. Factorial Formulas Watch out for double-negatives. Also can graph the parabola in your calculator and look for where the symmetry is The numerator is the total number of letters, the denominator backs out the repeats 23. Solving Inequalities 24. Graphs of trig functions Amplitude Frequency 25. Binomial Expansion Formula 26. Negative exponents 27. Case II Trinomial Factoring Just memorize it... I promise you want toβ¦ itβs a regentβs favorite Work ββ2 2 Isolate: cos π = (1) π = 45° (2) Quads 2 & 3 (3) 45° in quad 2 = 135° 45° in quad 3 = 225° β(β8) 8 π₯ = 2(4) = 8 = 1 Total: 10 letters A: 2 R: 2 10! 2!2! Isolate the absolute value part Think whatβs inside the absolute value brackets can be equal to (because the absolute value of 8 is 8), or it could be equal to -8 (because the absolute value of -8 is 8) Sine goes through origin; cosine doesnβt Amplitude tells you how high the graph goes Frequency tells you how many FULL curves happen in 360° Also... plug the options into your calculator and see which one matches! (Zoom 7: ZTrig) For the π π‘β term of (π + π)π ( π πΆπβ1 )(ππβ(πβ1) )(ππβ1 ) |3π₯ β 4| = 8 3π₯ β 4 = 8 π₯=4 Move all of the negative exponents to the other side of the fraction line first, then combine and/or simplify as much as possible Look for factors that multiply to -24 (-8*3) and add to -10. Rewrite as 4 terms, then factor by grouping. 20π₯ 5 35π₯ 3 π¦ 2 π¦ 3 20π₯ 5 4π₯ 2 = 35π₯ 3 π¦ 5 7π¦ 5 3π₯ β 4 = β8 4 π₯ = β3 π = 6, π = 4, π β 1 = 3 ( 6 πΆ3 )(π₯ 6β3 )(43 ) (20)(π₯ 3 )(64) 1280π₯ 3 3π₯ 2 β 12π₯ + 2π₯ β 8 3π₯(π₯ β 4) + 2(π₯ β 4) (3π₯ + 2)(π₯ β 4) Topics to Know PRACTICE TEST #4 β FULL ANALYSIS Explanation 28. Solving Radical Expressions Isolate the radical first Square both sides to get rid of the radical 29. 30. Compliments of Cofunctions are Equal THE TWO ANGLES ARE NOT EQUAL (Reference Sheet Formula) Sin and cos are COfunctions They are equal SO, the two angles must be compliments (add to 90°) 2 sides 2 anglesβ¦ Law of SINES 31. Circle Formula 32. Solving Rational Expressions Center... opposite sign Radiusβ¦ square root Find LCD Multiply each term by whatβs MISSING Solve Work β3π₯ β 5 = 2 3π₯ β 5 = 4 π₯=3 3π₯ β 4 + 5π₯ + 14 = 90 8π₯ + 10 = 90 π₯ = 10 23 sin 61 π = sin 40 π = 16.9 Center (-15, -1) Radius = β196 = 14 10 4 2 + 3 = π₯β3 LCD: (π₯ β 3)(3) π₯β3 3 10 π₯β3 4 3 2 (3) π₯β3 + (π₯β3) 3 = (3) π₯β3 30 3(π₯β3) 4π₯+18 3(π₯β3) 4π₯β12 6 + 3(π₯β3) = 3(π₯β3) = 6 3(π₯β3) 4π₯ + 18 = 6 π₯ = β3 33. Rationalizing Denominators Multiply top and bottom by the conjugate. Simplify carefully 34. Sigma Plug in all integers from the bottom number (4) to the top numbers (7) into the expression given. Add up all of the answers 35. Plug in π is a number, not a variable 26 7ββ5 β 7+β5 7ββ5 182β26β5 182β26β5 2(91β13β5) 91β13β5 = = = 49β5 44 44 22 π₯ π₯ π₯ π₯ =4 =5 =6 =7 5(4) β 12 = 8 5(5) β 12 = 13 5(6) β 12 = 18 5(7) β 12 = 23 8 + 13 + 18 + 23 = 62 π = (5200)(π 0.041β12 ) $8,505.04 Topics to Know 36a Quadratic Formula PRACTICE TEST #4 β FULL ANALYSIS Explanation Carefully go through the arithmetic & simplifying 36b Factor by Grouping Split into two groups of two, GCF Factor each group. The leftover binomial should match! 37a Input to L1 and L2. Stat Calc LinReg Calculator Statistics 37b Binomial Probability 38a Magnitude setup (parallelogram & triangle) Reference Sheet Formula 38b Reference Sheet Formula ( π πΆπ )(ππ )(π πβπ ) n=total, r=want, p=prob of success, q=prob of failure Success raised to success.. failure raised to failure Setup the parallelogram first, consecutive angles are supplementary. Label everything you know. Youβre solving for the diagonal through the middle. Pull out & relabel a triangle 3 sides, 1 angleβ¦ Law of COSINES 2 sides 2 anglesβ¦ Law of SINES Work β(β6) ± β(β6)2 β 4(1)(34) 2(1) 6 ± β36 β 136 π₯= 2 6 ± ββ100 π₯= 2 6 ± 10π π₯= 2 π₯ = 3 ± 5π 2π₯ 3 β 10π₯ 2 β 18π₯ + 90 = 0 2π₯ 2 (π₯ β 5) β 18(π₯ β 5) = 0 (2π₯ 2 β 18) (π₯ β 5) = 0 2π₯ 2 β 18 = 0 π₯β5=0 π₯ = β3 π₯ = 3 π₯ = 5 a) π¦ = 0.774π₯ + 19.093 b) π = 0.9045 (Make sure diagnostics are ON) c) π¦ = 0.774(12) + 19.093 π¦ = $28 βAt least 5β means 5 OR 6 5 games won: ( 6 πΆ5 )(. 635 )(. 371 ) = 0.2203 6 games won: ( 6 πΆ6 )(. 636 )(. 370 ) = 0.0625 βORβ means add them togetherβ¦ 0.2828 π₯= π₯ 2 = 422 + 652 β 2(42)(65)(cos 140) 16.4 sin 68 12.7 = sin π 12.7βsin 68 sin π = = 0.7180 16.4 β1 π = sin 0.7180 = 46° Topics to Know 39 Review packet, Trig page, bottom left cornerβ¦ SO IMPORTANT!!! PRACTICE TEST #4 β FULL ANALYSIS Explanation Isolate (this one is caseI factoring)β¦ ref angleβ¦ quadrantsβ¦ put it all together Work 2 tan π β 4 tan π β 12 = 0 (tan π β 6)(tan π + 2) = 0 tan π β 6 = 0 tan π = 6 (1) π = tanβ1 6 π = 81° (2) Positive. Quads 1 & 3 (3) 81° in quad 1 = 81° 81° in quad 3 = 261° tan π + 2 = 0 tan π = β2 (1) π = tanβ1 2 π = 63° (2) Negative. Quads 2 & 4 (3) 63° in quad 2 = 117° 63° in quad 4 = 297° π½ = {ππ°, πππ°, πππ°, πππ°}