Released Items 2014 ANS

Fall 2014 NC Final Exam: Precalculus Question: What transformations have occurred to create the function 𝑓 ( 𝑥 ) = 3 𝑥 3 − 4 f(x)=3x 3 −4 from the function 𝑔 ( 𝑥 ) = 𝑥 3 g(x)=x 3 ? Answer: D) The graph of the function has been stretched vertically and shifted down four units. Question: An object is launched straight upward from ground level with an initial velocity of 50.0 feet per second. The height, ℎ h (in feet above ground level), of the object 𝑡 t seconds after the launch is given by the function ℎ ( 𝑡 ) = − 16 𝑡 2 + 50 𝑡 h(t)=−16t 2 +50t. At approximately what value of 𝑡 t will the object have a height of 28.0 feet and be traveling downward? Answer: A) 2.39 seconds Question: What is the range of the function 𝑓 ( 𝑥 ) = − 5 ( 𝑥 + 3 ) 2 − 5 f(x)=−5(x+3) 2 −5? Answer: C) ( − ∞ , − 5 ] (−∞,−5] Question: A wind is blowing from the northwest toward the southeast. If the eastward component has a magnitude of 5.00 miles per hour and the southward component has a magnitude of 15.00 miles per hour, in what direction is the wind blowing? Answer: D) The wind is blowing in the direction 18.4° east of south. Question: What value of 𝑥 x satisfies the equation log ⁡ 3 ( 𝑥 − 4 ) = 2 log 3 ​ (x−4)=2? Answer: D) 13 Question: A man is standing on level ground 50 feet away from the wall of a building. He looks up at a window on the building. The angle of elevation to the bottom of the window is 28.5°. He then looks up at the top of the building. The angle of elevation to the top of the building is 35°. What is the approximate distance elevation between the bottom of the window and the top of the building? Answer: B) 7.9 feet Question: Triangle WXY has the following properties: Angle at vertex W is 149°. Side opposite vertex W has a length of 7.00 units. Side opposite vertex X has a length of 9.00 units. What is the approximate length of the side opposite vertex Y? Answer: B) 2.08 units Question: How should the graph of 𝑓 f be shifted to produce the graph of 𝑔 g? 𝑓 ( 𝑥 ) = 3 sin ⁡ ( 2 𝑥 ) + 4 f(x)=3sin(2x)+4 𝑔 ( 𝑥 ) = 3 sin ⁡ ( 2 𝑥 − 𝜋 4 ) + 4 g(x)=3sin(2x− 4 π ​ )+4 Answer: C) Shift the graph of 𝑓 f to the left 𝜋 8 8 π ​ units. Question: The maximum height, in inches, a ball reaches after its first four bounces is shown in the table below: Bounce Number Height (in inches) 1 42.0 2 31.5 3 23.6 4 17.7 Which type of function best models the data and why? Answer: A) An exponential function, because the height of the ball is decreasing by 25% with each bounce. Question: What is the inverse function of 𝑔 ( 𝑥 ) = 𝑥 3 − 2 g(x)=x 3 −2? Answer: A) 𝑔 − 1 ( 𝑥 ) = 𝑥 + 2 3 g −1 (x)= 3 x+2 ​ Question: What are the polar coordinates of the point ( − 2 3 , 2 ) (−2 3 ​ ,2), where 0 ≤ 𝜃 ≤ 36 0 ∘ 0≤θ≤360 ∘ ? Answer: B) ( 2 , 13 5 ∘ ) (2,135 ∘ ) and ( 2 6 , 31 5 ∘ ) (2 6 ​ ,315 ∘ ) Question: Which equation is the rectangular form of the polar equation 𝑟 = 2 1 + cos ⁡ 𝜃 r= 1+cosθ 2 ​ ? Answer: C) 𝑦 2 + 4 𝑥 = 4 y 2 +4x=4 Question: Two parametric equations are shown below, where 𝑡 ≥ 0 t≥0. 𝑥 = 1 3 𝑡 + 3 − 7 x= 3 1 ​ t+3 ​ −7 𝑦 = 4 𝑡 2 − 7 y=4t 2 −7 Which non-parametric equation can be used to graph the curve described by the parametric equations? Answer: D) 𝑦 = 324 ( 𝑥 − 3 ) 4 − 7 y=324(x−3) 4 −7 Question: The formula for a sequence is shown below: 𝑎 1 = 2 , 𝑎 2 = 4 , 𝑎 3 = 3 , 𝑎 𝑛 = 3 a 1 ​ =2,a 2 ​ =4,a 3 ​ =3,a n ​ =3. Which is another formula that represents the sequence? Answer: A) 𝑓 ( 𝑛 ) = 3 ( 2 𝑛 − 1 ) f(n)=3(2 n−1 ) Question: When 𝑎 1 = 25 , 000 a 1 ​ =25,000, what is the sum of the infinite sequence defined by the equation 𝑎 𝑛 + 1 = 0.8 𝑎 𝑛 a n+1 ​ =0.8a n ​ ? Answer: A) 125,000 Question: What is the end behavior of the function 𝑓 ( 𝑥 ) = 100 1 + 5 ( 0.75 ) 𝑥 f(x)= 1+5(0.75) x 100 ​ ? Answer: B) lim ⁡ 𝑥 → ∞ 𝑓 ( 𝑥 ) = 0 lim x→∞ ​ f(x)=0 and lim ⁡ 𝑥 → − ∞ 𝑓 ( 𝑥 ) = 100 lim x→−∞ ​ f(x)=100. Question: In the piecewise function below, k is a constant: 𝑓 ( 𝑥 ) = 𝑥 2 − 𝑘 2 𝑥 − 𝑘 + 𝑘 f(x)= x−k x 2 −k 2 ​ +k for 𝑥 ≠ 𝑘 x  =k 4 − 𝑘 , 𝑥 = 𝑘 4−k,x=k What is the value of the limit lim ⁡ 𝑥 → 𝑘 𝑓 ( 𝑥 ) lim x→k ​ f(x)? Answer: B) 2 𝑘 2k