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Pre-Calculus 20 Final Review June 2015 Name: ____________________________ P20.1 Demonstrate understanding of the absolute value of real numbers and equations and functions involving the absolute value of linear and quadratic functions Level 2 1. Evaluate. Level 3 2. Solve the following equation, and verify .Label the extraneous solution if possible. (Level 3) Case 1: Case 2: |7π₯ β 3| = π₯ + 1 Verify: Verify: Level 4 3. A machine fills containers with 64 ounces of oatmeal. After the containers are filled, another machine weighs them. If the containerβs weight differs from the desired 64-ounces weight by more than ±0.8 ounces, the container is rejected. Write an absolute value equation that can be used to find the heaviest and lightest acceptable weights for the container. P20.2 Expand and demonstrate understanding of radicals with numerical and variable radicands Level 2 1.Express each entire radical as a mixed radical in simplest form. 2. Write each mixed radical as an entire radical. Level 3 3.Solve the following radical equations, and verify. Label the extraneous solution if possible. Level 4 4. The speed, V, in feet per second of outlow of a liquid from an orfice is given by the formula π = 8ββ , where h is the height, in eet, of the liquid above the opening. How high, to the nearest hundereth of a foot, is a liquid above an office if the velocity of outflow is 100 ft/s? P20.3 Expand and demonstrate understanding of rational expressions and equations Level 2 1. Simplify and state the non-permissible values for the variables. a) b) Level 3 2. Solve the following rational equation, and verify. Label the extraneous solution if possible. Level 4 3. Three numbers are related to one another such a way that the second number is 8 larger than the first and the third number is three times as large as the first. Find the numbers if the sum of three-quarters of the first number, two-thirds of the second number and one-sixth of the third number is 36. P20.4 Expand and demonstrate understanding of the primary trigonometric ratios including the use of reference angles and the determination of exact values for trigonometric ratios Level 2 1. Sketch each angle in standard position. State which quadrant the angle terminates in and the measure of the reference angle. a) 20° quadrant _________________________ ΞΈR= ____________________ b) 330° quadrant _________________________ ΞΈR= ____________________ Level 3 2. Solve for ΞΈ satisfy the equation for 0°<ΞΈβ€360°? 1 a)cos π = 0.3420 b) sin π = β 2 Level 4 3. A guy wire supporting a radio tower is positioned 135 feet up the tower. It forms a 45Λ angle with the ground. About how long is the wire, using the exact length? (Level 4) P20.5 Demonstrate understanding of the cosine law and sine law, including the ambiguous case Level 2 1. Find the indicated angle or side length. Round your answer to one decimal place. a) b) c) Level 3 2. In βπ΄π΅πΆ, where β π΄ = 29°, π = 17, and π = 18. Find angle(s) β πΆ. Level 4 3. A 5m flag pole is not standing up straight. There is a wire attached to the top of the pole and anchored in the ground. The wire is 5.17m long. The wire makes a 52° angle with the ground. What angle does the flag pole make with the wire? Round your answer to the nearest degree. P20.6 Expand and demonstrate understanding of factoring polynomial expressions Level 2 1. Factor the following: 1 a) π2 β 49π 2 b) π₯ 2 β 5π₯ β 6 c) 4π₯ 2 β 36π¦ 2 d) 6π₯ 2 + 5π₯ β 6 e) π₯ 2 β 3π₯ β 4 f) 2π₯ 2 + 7π₯ + 3 9 Level 3 2. Factor the following: Level 4 3. One side of an envelope is 5 inches longer than the other side. The area of the envelope is 110 in2. Determine the dimensions of the envelope. P20.7 Demonstrate understanding of quadratic functions of the form π¦ = ππ₯ 2 + ππ₯ + π and their graphs Level 2 1. a) Graph the following quadratic function: π¦ = β6(π₯ β 4)2 + 3 b) For the graph state the following: direction of opening: __________________________________________________ domain: _____________________________________________________________ range: _______________________________________________________________ axis of symmetry: ______________________________________________________ x-intercepts: _________________________________________________________ y-intercepts: __________________________________________________________ Level 3 2. a) Convert the following quadratic functions from standard form to vertex graphing form. b)State the vertex. Level 4 3. A basketball is thrown into the air from ground level and its path ia a parabola. It reaches a maximum height of 8 m and lands 20m from where it was thrown. Determine an eqation that models the path of the ball. P20.8 Demonstrate understanding of quadratic equations Level 2 1. Solve using any method. Level 3 2. Solve by completing the square. Express answers to two decimal places. Level 4 3. List the one advantage and one disadvantage to each way of solving a quadratic equation. Factoring: Completing the square Quadratic Formula: P20.10 Demonstrate understanding of arithmetic and geometric sequences and series Level 2 1. Identify whether the following sequences are arithmetic, geometric, or neither. Level 3 2. Find the sum of the first 10 terms of the arithmetic series. 6+3+0+β― Level 4 3. The length of an initial swing of a pendulum is 120cm. Each successive swing is 0.90 times the length of the previous swing. If this process continues forever, how far will the pendulum swing in total? P20.11 Demonstrate understanding of reciprocal functions Level 2 1 1. Use the graph of y= f(x) given here to graph π¦ = π(π₯) on the same axis. Show the vertical asymptotes on your graph. a) b) Level 3 1 2. Sketch the graph π¦ = 2π₯+2 . Label the asymptote(s). Level 4 3. Consider this graph of a reciprocal function π¦ = graph of the original function π¦ = π(π₯). 1 . On the same set of axes, sketch the π(π₯) Formulas π π π = = sin π΄ sin π΅ sin πΆ π 2 = π2 + π 2 β 2ππ cos πΆ βπ ± βπ 2 β 4ππ π₯= 2π Arithmetic π‘π = π‘1 + (π β 1)π π ππ = 2 [2π‘1 + (π β 1)π] π ππ = 2 (π‘1 + π‘π ) Geometric π‘π = π‘1 (π)πβ1 ππ = π‘1 (π π β 1) (π β 1) ππ = ππ‘π β π‘1 πβ1 π‘ 1 πβ = (1βπ)