Graphing Trigonometric Skills


 Amanda Bradford
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1 Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE BELOW! All problems should be done in order. If you need to skip a problem, leave room for it and go back later to complete it. page # problem # # problem # AP Calculus Prerequisite Skills Packet Note: Problem numbers should be to the left of the pink margin, and everything else should be to the right. This makes it much easier for me to spotcheck your work and for you to find problems quickly. This packet includes a sampling of problems that students entering an AP Calculus course should be able to answer. The questions are organized by topic: A Basic Algebra Skills L Linear Functions FE Functions & Equations Q Quadratic Functions EL Eponential and Logarithmic Functions R Rational Epressions and Equations G T Graphing Trigonometric Skills Name Date Show work clearly (you don t have to copy the problem) BOX your answer Students entering AP Calculus absolutely must have a strong foundation in algebra. Some questions in this packet were included because they concern skills and concepts that will be used etensively in AP Calculus, whereas others have been included because being able to answer them indicates a strong grasp of important mathematical concepts and more importantly the ability to think and problemsolve. On the first day of class, we will go over your work. I will spotcheck it for completion and / or accuracy. Remember, completion includes SHOWING WORK. Our first test will cover this material.
2 A: Basic Algebra Skills A. True or false. If false, change what is underlined to make the statement true. 4 a. T F b. T F ( + ) = + 9 T F d. T F e. (4 + ) = 6( + ) T F f. 5 T F g. If ( + )( 0) =, then + = or 0 =. T F A. More basic algebra. a. If 6 is a zero of f, then is a solution of f () = 0. b. Lucy has the equation (4 + 6) 8 = 6. She multiplies both sides by ½. If she does this correctly, what is the resulting equation? Simplify 4 0 d. Rationalize the denominator of e. If f () = + + 5, then f ( + h) f () = (Give your answer in simplest form.) f. A cone s volume V is given by V r h. If the radius r is three times the height h, write V in terms of h. g. Write an epression for the area of an equilateral triangle with side length s. h. Suppose an isosceles right triangle has hypotenuse h. Write an epression, in simplest form, for its perimeter in terms of h.
3 L: Linear Functions L. One of the following equations is in standard form, one is in pointslope form, and one is in slopeintercept form. For each problem, write its letter in the appropriate column, find the requested information, and graph the equation. (A) y 4 = ( + ) (B) 4y 6 (C) y 5 Standard PointSlope SlopeIntercept ( ) ( ) ( ) intercept: point on line: slope: yintercept: slope: yintercept: L. Write the equation of each line described in pointslope and slopeintercept form. PointSlope SlopeIntercept a. slope: / through: (6, ) b. through (4, ) and (5, 6) through (0, ) perp. to 4 + y = 7 d. through (6, ) parallel to y = ( + ) L. Word Problems / Applications. a. Sweatshirts sell for $5 each, and tshirts sell for $5 each. A store made $45 on sweatshirt and tshirt sales. Let S be the number of sweatshirts sold, and let T be the number of tshirts sold. Write an equation in standard form for this situation. b. At 40 C, sound travels at a rate of 55 m/s. For each increase in temperature, sound travels ½ m/s faster. Let T be the temperature, and let S be the speed of sound. Write an equation in pointslope form for this situation.
4 FE: Functions & Equations FE: For each diagram: i. Decide if the relation is a function ii. State the domain and range iii. Determine if the function is onetoone **Assume that if a graph does not have end points, that it continues in that direction infinitely.** a. b. d. e. f. g. h. i. FE: Sketch a graph of the function and identify any horizontal and/or vertical asymptotes. a. y 6 b. y y 9 FE: Given that f, g, and h a. g h b. g f 6 h, find d. f g FE4: The graph shows the function f(), for 4. h f. Sketch the graph of h(). a. Let. The point A(, ) on the graph of f is transformed to the point P on the graph of g. Find the coordinates of P. b. Let g f FE5: Consider the functions f and g where f and g Let a. Find the inverse function, f b. Given that g, find g f Show that f g h, f g. d. Sketch the graph of h for 6 0 and 4 y 0, including any asymptotes. e. Write down the equations of the asymptotes.
5 Q: Solve each equation. a. 6 b. d Q: Quadratic Functions e. f Q: Write an equation to represent each parabola below. a. b. Q: Determining the number of solutions of a quadratic function. a. The equation k 0 has two equal real roots. Find the possible values of k. b. Find the value of p such that the equation p 0 has two different real roots. Find the values of m such that the equation f 5 Q4: Let m 8 0 has no real roots. a. Write the function of f, giving your answer in the form f a h k. b. The graph of g is formed by translating the graph of f by 4 units in the positive direction and 8 units in the positive ydirection. Find the coordinates of the verte of the graph of g. Q5: The height of a ball t seconds after it is thrown is modeled by the function where h is the height of the ball in meters. a. Find the maimum height reached by the ball. b. For what length of time will the ball be higher than meters? h t t 5 4.9, Q6: A piece of wire 40 cm long is cut into two pieces. The two pieces are formed into two squares. a. If the side length of one of the squares is cm, what is the side length of the other square? b. Write an equation for the combined area of the two squares. What is the minimum combined area of the two squares? Q7: The sum of the squares of three consecutive positive odd integers is 5. Find the integers. Q8: A homebuilder wants to build a rectangular deck on the back of a house. One side of the deck will share a wall with the house, and the other three sides will have a wooden railing. If the builder has enough wood for 5 meters of railing, what is the area of the largest deck he could build? Q9: Jaswinder takes a trip to visit his sister, who lives 500 km away. He travels 60 km by bus, and 40 km by train. The train averages 0 km/h faster than the bus. If the entire journey takes 8 hours, find the average speeds of the bus and the train.
6 EL Eponential & Logarithmic Functions EL: Simplify each epression as much as possible so that each eponent is a positive integer. 5 4y a. b. y 6 q q 64a d..5 y e. 7c d f. 6y 8 g h. q a b a b EL: Solve these equations for, without the use of a calculator. a. 9 b d. 9 EL: Evaluate these epressions 4 a. log b. log 64 log d. ln e. ln 4 ln 5 EL4: Solve these equations. a. log 4 d. b. log log e e f. 5e 0.5 EL5: Solve the equation 6 4 b are positive real numbers. to find the value of in the form ln a, where a and ln b EL6: The sum of 450 is invested at.% interest, compounded annually. a. Write down a formula for the value of the investment after n years. b. After how many years will the value first eceed 600? EL7: Joseph did a parachute jump for charity. After jumping out of the aircraft his velocity at time t 0.06t seconds after his parachute opened was v m/s where v9 9e. a. Sketch the graph of v against t. b. What was Joseph s speed at the instant the parachute opened? What was his lowest possible speed if he fell from a very great height? d. If he actually landed after 45 seconds what was his speed on landing? e. How long did it take him to reach half the speed he had when the parachute opened. EL8: Two variable and n are connected by the formula n =, = 08. Find the values of a and b. EL9: Given that otherwise solve b a n. When n =, = and when log log 7 log A find an epression for A in terms of. Hence or log log 7. EL0: Solve. For each, give an eact answer and an answer rounded to 4 decimal places. a. ln ( + ) = b. ln + ln 4 = ln + ln ( + ) = ln d. ln ( + ) ln ( ) = ln
7 R. Function a. f( ) 4 b. R Rational Functions Hole(s): (, y) Domain if any Horiz. Asym., if any (4 ) 48 f ( ) 6 4 f ( ) skip skip 0 8 Vert. Asym.(s), if any R: Write the equation of a function that has a. asymptotes y = 4 and =, and a hole at (, 5) b. holes at (, ) and (, ), an asymptote = 0, and no horizontal asymptote R: Find the coordinates where the function s output is zero and where it is undefined. Then sketch a graph of the function. 6 a. f( ) 5 b. g ( ) R4: Simplify completely. a. b. d ( ) ( ) (Don t worry about rationalizing) (Your final answer should have just one numerator and one denominator) (Don t worry about rationalizing) R5: People with sensitive skin must be careful about the amount of time spent in direct sunlight. The.s 48 relation m where m is the time in minutes and s is the sun scale value, gives the s maimum amount of time that a person with sensitive skin can spend in direct sunlight without skin damage. a. Sketch this relation when 0 s 0 and 0 m 00 b. Find the number of minutes that skin can be eposed when i. s = 0 ii. s = 40 iv. s = 00 What is the horizontal asymptote? d. Eplain what this represents for a person with sensitive skin.
8 The Unit Circle MUST be memorized. Angles should be known in both degrees and radians. = (, ) = (, = (, = (, ( ) = 0 = (, = (, = (, = (, (, ) =
9 You should know the graphs of these si functions: Sine Cosine Tangent y = sin y = cos y = tan Domain: D: D: Range: R: R: D: R: Cosecant Secant Cotangent y = csc y = sec y = cot D: D: R: R: tan Trig Identities & Formulas (also on the insidefront cover of your book) The identities with boes around them must be memorized. sin cos csc Reciprocal Identities sec sin cos cot tan cos sin Pythagorean Identities sin cos tan sec cot csc Sum and Difference Formulas sin( y ) sin cosy cos sin y cos( y ) cos cosy sin sin y DoubleAngle Formulas sin( ) sin cos cos( ) cos sin cos sin sin cos sin cos csc sec HalfAngle Formulas cos cos tan Cofunction Identities cos sin sec csc cos cos (Use quadrant to determine sign.) tan cot cot tan
10 T: Trigonometry You should be able to answer these quickly, without referring to (or drawing) a unit circle. You will have pop quizzes in class with questions like these, especially T and T. T. Find the value of each epression, in eact form. a. sin b. cos 6 4 tan d. sec 5 7 e. 4 csc f. cot 5 6 T. Find the value(s) of in [0, ) which solve each equation. a. sin b. cos tan d. sec e. csc is undefined f. cot T. Solve the equation. Give all real solutions, if any. a. sin b. cos( ) tan 0 d. 4 sec 9 e. csc( 4 ) 0 f. cot6 0 T4. Solve by factoring. Give solutions in [0, ) only. a. 4sin + 4 sin + = 0 b. cos cos = 0 sin cos sin = 0 d. tan + tan = +
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