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Example Items Pre-Calculus Pre-Calculus Example Items are a representative set of items for the ACP. Teachers may use this set of items along with the test blueprint as guides to prepare students for the ACP. On the last page, the correct answer and content SE is listed. The specific part of an SE that an Example Item measures is NOT necessarily the only part of the SE that is assessed on the ACP. None of these Example Items will appear on the ACP. Teachers may provide feedback regarding Example Items. (1) Download the Example Feedback Form and email it. The form is located on the homepage of Assessment.dallasisd.org. OR (2) To submit directly, click “Example Feedback” after you login to the Assessment website. Second Semester 2016–2017 Code #: 1121 ACP Formulas Pre-Calculus/Pre-Calculus PAP 2016–2017 Trigonometric Functions and Identities Pythagorean Theorem: a2 + b2 = c2 Special Right Triangles: 30° - 60° - 90° x, x 3, 2x 45° - 45° - 90° x, x, x 2 Law of Sines: sin A sin B sin C = = a b c Heron’s Formula: A= Law of Cosines: a2 = b2 + c2 – 2bc cos A b2 = a2 + c2 – 2ac cos B c2 = a2 + b2 – 2ab cos C Linear Speed: v = Angular Speed: ω s t sin θ = Reciprocal Identities: 1 csc θ 1 csc θ = Pythagorean Identities: Sum & Difference Identities: Double-Angle Identities: cos θ = sec θ = sin θ sin2 θ + cos2 θ = 1 1 s ( s − a) ( s − b ) ( s − c ) θ = t tan θ = sec θ 1 cot θ = cos θ 1 + tan2 θ = sec2 θ 1 cot θ 1 tan θ 1 + cot2 θ = csc2 θ cos( α + β ) = cos α cos β − sin α sin β sin(α + β ) = sin α cos β + cos α sin β cos(α − β ) = cos α cos β + sin α sin β sin(α − β ) = sin α cos β − cos α sin β sin2x = 2 sin x cos x cos 2 x = cos2 x − sin2 x cos 2x = 2 cos2 x − 1 cos 2x = 1 − 2 sin2 x Projectile Motion 1 2 gt + (v0 sin θ )t + y0 2 Vertical Position: y =− Vertical Free-Fall Motion: s(t ) = − 1 2 gt + v0t + s0 2 Horizontal Distance: x = (v0 cos θ )t v(t ) = − gt + v0 g ≈ 32 ft m ≈ 9.8 sec2 sec2 Conic Sections Parabola: (x - h)2 = 4p(y - k) (y - k)2 = 4p(x - h) Circle: x2 + y2 = r2 (x – h)2 + (y - k)2 = r2 Ellipse: ( x − h) Hyperbola: ( x − h) 2 a2 + 2 a2 − (y − k ) 2 (y − k ) b2 2 =1 b2 ( x − h) 2 + b2 (y − k ) 2 =1 a2 − (y − k ) 2 =1 a2 ( x − h) b2 2 =1 ACP Formulas Pre-Calculus/Pre-Calculus PAP 2016–2017 Exponential Functions Simple Interest: I = prt Compound Interest: r A = P 1 + n Exponential Growth or Decay: N = N0 (1 + r ) nt t Continuous Compound Interest: A = Pert Continuous Exponential Growth or Decay: N = N0ekt Sequences and Series The nth Term of an Arithmetic Sequence: an = a1 + (n − 1)d Sum of a Finite Arithmetic Series: a Sum of a Finite Geometric Series: a Sum of an Infinite Geometric Series: a Binomial Theorem: (a + b) Permutations: n n k =1 k k =1 k ∞ n =1 n Pr = an = a1r n−1 n (a + an ) 2 1 = n The nth Term of a Geometric Sequence: = a1(1 − r n ) , r ≠1 1−r = a1 , r ≠1 1−r n Sn = a1 − an r , r ≠1 1−r = n C 0 an b0 + n C1 an −1 b1 + n C2 an − 2 b2 + ⋅ ⋅ ⋅ + n C n a0 b n n! (n − r )! Combinations: n Cr = n! (n − r )! r ! Coordinate Geometry Distance Formula: d = ( x2 − x1 )2 + (y2 − y1 )2 Slope of a Line: m= Midpoint Formula: x + x2 M= 1 , 2 Quadratic Equation: ax2 + bx + c = 0 y2 − y1 x2 − x1 y1 + y2 2 Quadratic Formula: Slope-Intercept Form of a Line: y = mx + b Point-Slope Form of a Line: y − y1 = m(x − x1 ) Standard Form of a Line: Ax + By = C x = −b ± b2 − 4ac 2a HIGH SCHOOL Page 1 of 8 EXAMPLE ITEMS Pre-Calculus, Sem 2 1 Which graph represents the curve given by the parametric equations x y 8 sin t over the interval 0 t 2 ? A C B D Dallas ISD - Example Items 3 cos t and Page 2 of 8 EXAMPLE ITEMS Pre-Calculus, Sem 2 2 3 4 What is the rectangular form for the curve given by the parametric equations x y 5t 3 ? A y 5x 33 B y 5x 3 C y 5x 27 D y 5x 9 t 6 and Which pair of parametric equations represents a line that passes through points (2, 1) and (0, –3)? A x y 2t 4t 3 B x y 2t 8t 3 C x y 2t 4t 3 D x 2t y 8t 3 5 What are the rectangular coordinates of the point 5, ? 6 A 5 3 5 , 2 2 B 5 5 3 , 2 2 C 5 5 3 , 2 2 D 5 3 5 , 2 2 Dallas ISD - Example Items Page 3 of 8 EXAMPLE ITEMS Pre-Calculus, Sem 2 5 A polar equation is used to produce the graph of the rose shown. Which equation is used to create the rose? 6 A r 2 sin(2 ) B r 2 sin(4 ) C r 3 sin(4 ) D r 3 sin(2 ) The intersection of a plane and a double-napped cone is shown in the diagram. What type of conic section is formed by this intersection? A Circle B Ellipse C Hyperbola D Parabola Dallas ISD - Example Items Page 4 of 8 EXAMPLE ITEMS Pre-Calculus, Sem 2 7 The graph of an ellipse is shown. Which equation represents this ellipse? A ( x 1)2 (y 2)2 3 5 1 B ( x 1)2 (y 2)2 3 5 1 C ( x 1)2 (y 2)2 9 25 1 D (x 1)2 (y 2)2 9 25 1 Dallas ISD - Example Items Page 5 of 8 EXAMPLE ITEMS Pre-Calculus, Sem 2 8 A hyperbola has foci at (–1, –2) and (13, –2) and an eccentricity of 7 . What is the equation of 6 the hyperbola in standard form? 9 A (x 6)2 (y 2)2 36 85 1 B (x 6)2 (y 2)2 36 85 1 C ( x 6)2 (y 2)2 36 13 1 D ( x 6)2 (y 2)2 36 13 1 What is the exact value of the trigonometric function cos(–870°)? A B C 1 2 D 10 3 2 1 2 3 2 What is the reference angle for an angle that measures A 7 4 B 5 4 C 3 4 D 4 Dallas ISD - Example Items 17 radians? 4 Page 6 of 8 EXAMPLE ITEMS Pre-Calculus, Sem 2 11 A cable holds an 80-foot pole straight upright, as shown. Based on the given information, what is the approximate length of the cable, to the nearest tenth of a foot? Record the answer and fill in the bubbles on the grid provided. Be sure to use the correct place value. 12 A helicopter is flying from downtown Dallas to downtown Fort Worth. The distance between the two cities is 32 miles. 45° 35° ? Dallas 32 miles Fort Worth If the angle of depression from the helicopter to Dallas is 45° and the angle of depression to Fort Worth is 35°, approximately how far is the helicopter from downtown Dallas? A 18.6 miles B 23.0 miles C 26.0 miles D 39.4 miles Dallas ISD - Example Items Page 7 of 8 EXAMPLE ITEMS Pre-Calculus, Sem 2 13 The diagram shows a boat that is anchored at point B in a river. There are two boat ramps on the far side of the river, shown by points A and C. The boat is 120 meters from ramp A and 150 meters from ramp C. If mABC meter? 110° , what is the approximate distance between the two boat ramps, to the nearest Record the answer and fill in the bubbles on the grid provided. Be sure to use the correct place value. 14 Harrison walks to the library after school every day. When Harrison leaves school, he walks 16 blocks due West and then 12 blocks due North to get to the library. What is the magnitude and direction of the resultant vector? A Magnitude: 20 blocks Direction: W 41.4° N B Magnitude: 20 blocks Direction: W 36.9° N C Magnitude: 28 blocks Direction: W 41.4° N D Magnitude: 28 blocks Direction: W 36.9° N Dallas ISD - Example Items Page 8 of 8 EXAMPLE ITEMS Pre-Calculus, Sem 2 15 16 If r = 4, –2, which graph represents –2r? A C B D If u = 8, 12, –3, v = –4, 7, 14, and w = 2, –5, 6, what is 3u – 4v + 2w? A 6, 14, 17 B 19, 24, 23 C 12, 54, 9 D 44, –2, –53 Dallas ISD - Example Items EXAMPLE ITEMS Pre-Calculus, Sem 2 Answer SE Process Standards 1 B P.3A P.1B, P.1D, P.1E, P.1F 2 A P.3B P.1B, P.1D, P.1E, P.1F 3 C P.3C P.1B, P.1D, P.1E, P.1F 4 A P.3D P.1B, P.1C, P.1D, P.1E, P.1F 5 D P.3E P.1B, P.1D, P.1E, P.1F 6 B P.3F P.1F 7 C P.3H P.1B, P.1E, P.1F 8 D P.3I P.1B, P.1D, P.1E, P.1F 9 A P.4A P.1B, P.1C, P.1E, P.1F 10 D P.4C P.1B, P.1C, P.1E, P.1F 11 90.6 P.4E P.1A, P.1B, P.1C, P.1F 12 A P.4G P.1A, P.1B, P.1C, P.1F 13 222 P.4H P.1A, P.1B, P.1C, P.1F 14 B P.4I P.1A, P.1B, P.1C, P.1F 15 C P.4J P.1B, P.1C, P.1E, P.1F 16 D P.4K P.1B, P.1C, P.1E, P.1F Dallas ISD - Example Items