A2TH MIDTERM REVIEW (at home) )Simplify 3 2[ 2x 3(x + 4) 3(4 x) ] CHAPTER /2 NUMBERS/FUNCTIONS 2) Solve the inequality, write the solution set, and graph in the specified domain: a) 5x 3< 22 {integers} b) 3 x 5 {reals} c) 3 x 2 < 5 {reals} 3) KNOW THE CHART ON P.3 IN YOUR TEXT!! (NUMBER SETS)- there is also an EDPUZZLE on this. 4) KNOW THE AXIOMS ON P. 4&5. 5) Circle the functions (more in section 2.4!) Draw a graph that represents the scenario (more in section 2.3) 6a) The temperature of your cup of coffee is related to how long it has been cooling 6b) When you dive off the 3-meter diving board, time, and your position in relation to the water s surface are related
7) Use the graph of f (in the middle) to sketch: y = f (x) + 2 y = f (x) y = f (x 2) y = 2f (x) 8) Graph the piecewise functions 3 x x < f (x) = x + 4 x < 5 5 x 5 f (x) = x 2 4 x < 2 (x 3) 3 + 3 x 2 Evaluate f(0) f(5) f(0) Evaluate f(2) f(0)
9) Find the domain and range in interval notation (more in section 2.2) 0) Find the domain and range in interval notation (check desmos!): f (x) = x 2 4 f (x) = x + 3 x f (x) = 2 3 x f (x) = 4 x + 4 ) Is the function EVEN, ODD, or NEITHER? (there is an EDPUZZLE on this) f (x) = x 2 + x 4 + 3 f (x) = 2x 3 + 5x f (x) = x 3 + x + 2) Function Operations (check my website- FUNCtions, Function Operations/Composition practice with answers! For 40 more of these ) a) Given f (x) = 4x 2 b) c) f (x) = x 3 + 3 f (x) = 4x + 5 d) g(x) = 3x + 3 Find f (x 2 ) Find ( f!f )(7) Find f (x) g(x) f (x) = 4x g(x) = x 3 e) Given f (x) = g(x) = x + 3, find (if possible), (f!g)(0) and (f!g)( 3) x Find(4f 4g)(9) f) Given f(x)= -(x+) 2 2, write a function, h(x), where h(x) = f(x) + 4 g) Given f(x)= 2(x-) 2 +, write a function, h(x), where h(x) = f(x-5) h) Given f(x)= 4(x-2) 2 + 3, write a function, h(x), where h(x) = 2f(x)
3) GRAPHING EXTRAVAGANZA!!! We graphed throughout the year, but I put everything here to make it flow better Please use transformations from the PARENT function. Do not choose random points, even if they are in the domain! Provide the domain and range in interval notation! SHOW ASYMPTOTES WHERE NECESSARY!!! f (x) = 2( x + 2) 2 3 f (x) = 3 x 5 y 5 = 2( x + 2) Domain: Range: Domain: Range: Domain: Range: f (x) = x 2 + 3 3 f (x) = x f (x) = 2 x 4 + Domain: Range: Domain: Range: Domain: Range:
f (x) = (x 4) 3 2 y + 2 = 2x 3 f (x) = 3x 2 + 6x + Domain: Range: Domain: Range: Domain: Range: x = y 2 f (x) = 2log 2 ( x) + 3 f (x) = 2 x 2 + x Domain: Range: Domain: Range: Domain: Range:
4) Graph f (x) = 9 in the domain 2 x 0. What is the range? x 5) Find the inverse function: f (x) = x 2 + 5 f (x) = x 3 2 f (x) = e x +2 f (x) = log 3 (x 2) 6) Is the function one-to-one? 2 f (x) = x f (x) = 2 x + 3 f (x) = x 2 + 3 f (x) = ln(x 2) + 4 7) Given f(x) and h(x), describe the transformations from f to h (use words like- reflections[be sure to specify which axis], vertical stretch/shrink, vertical/horizontal translation) f (x) = x 2 h(x) = 2(x 2) 2 3 f (x) = x h(x) = x 3 f (x) = e x h(x) = 2e x f (x) = lnx h(x) = 2 ln(x 2
CHAPTER 3 / 4 LINEAR FUNCTIONS/SYSTEMS/ MATRICES ) How do you find the X-INTERCEPT of a function? Y- INTERCEPT? 2) Find the equation of the line, in slope-intercept form, containing the points (-4,5) and (6,0) 3) Graph: x=3 y=-2 4) Find a slope-intercept form equation that contains the point (7,2) and is parallel to the graph of y= -4x + 3 5) Write the slope-intercept form of the equation of the function that is perpendicular to the graph of 5x 7y = 44 and contains the point (4,)
YOU CAN USE A CALCULATOR FOR THESE 2 6) Suppose you own a car that is presently 40 months old. The Blue Book value of the car claims that the present trade-in value is $3300. From and old Blue Book, you find that the trade in value 0 months ago was $4700. Assume that its trade-in value decreases linearly with time. a) Write the particular equation expressing trade-in value (v) of your car as a function of its age in months (a). b) You plan to get rid of the car when its trade-in value drops to $000. How much longer can you keep the car? c) By how many dollars does the car depreciate (decrease in value) each month? What part of your mathematical model tells you this? d) Find and interpret the a- and v- intercepts of your model. e) The car actually cost $0,560 when it was brand new. How do you explain the difference between this and what the trade in value was when it was brand new? 7) The speed of a bullet is traveling depends on the number of feet the bullet has traveled since it left the gun. Assume number of feet. s = 4d + 3600 a) How fast is the bullet going when it has traveled 300 ft?, where s is the number of feet per second and d is the b) How far has the bullet gone when it has slowed to 500 feet per second? c) What does the slope represent? d) What does the d-intercept equal? What does it represent? e) Write a suitable domain for the linear function.
8) Classify the system as inconsistent, dependent, or independent (check out the visual on p. 6 for help with this. You can check slopes and y-intercepts OR use elimination/substitution for this) 2x + 3y = 5 6x + 9y = 5 x + 2y = 7 2x 3y = 5 x + 2y = 3 2y = x 9) Solve the system algebraically (no calculator) (check with a calc for practice) (more of these on my website- Linear Functions 3-variable systems practice) x + 2y + z = 0 3x + 2y z = 4 x + 2y + 3z = 4 0) Solve. State any extraneous solutions: 2 x 5 y = 5 3 x + 0 y = 8 x 2 = 6y 64 x = 6 y x 2 + y 2 = 2 x + y = 2 ) You have 22 coins in your pocket, all quarters and dimes. They total to $4.5. How many of each type do you have?
2) You arrive late to school because your car broke down (you stopped at Starbucks ;] ) and you missed part of your exam. Your teacher, Mr. Gleeman is only going to give you the 45 minutes remaining in class to finish the exam, because you did not bring him a venti Emperor s Clouds and Mist Green Tea (no milk or sugar). Hint Hint. Jk. Sort of. The exam has 2 open-ended questions and 30 multiple-choice questions. You know that it usually takes you 5 minutes to answer an open-ended question and minute to answer a multiple choice question. Constraints: Each correct open-ended question is worth 20 points, and each multiple choice question is worth 2 points. Objective Function: Graph: How many of each type of question should you answer to maximize your score? What is the maximum score you can get given these constraints?
3) Multiply (if possible) (the columns in the first have to match the rows in the second!) 2 4 3 6 0 5 0 4 2 0 2 3 2 3 The determinant of matrix a b a b c d is c d = ad cb The determinant of a 3x3 matrix is: 4) Find the determinant: 3 2 5 3 2 0 4 3 2 0 3 2 x 5) The determinant of the matrix is -2. 3 2 What is x? 3 2 3
CHAPTER 5 QUADRATICS/IMAGINARY NUMBERS *For graphing quadratics- see graphing section above **We spent some time doing the Fundamental Theorem of Algebra, factoring, and solving here, but I included that in the Polynomials review. ) Rewrite f (x) = 2 x 2 + 4x 3 in vertex form by completing the square 2) Find the discriminant. Use it to reason about the nature of the roots (aka- are there two real solutions, one real solution with multiplicity, or no real solution?) 0 = 5x 2 +x + 2 2 = 9 = 6x + x 2 2 x 2 + 5x 3) What do the x-intercepts of the graph of a quadratic represent? 4) Solve by completing the square 2x 2 + 5x += 0 5) What are the intervals of increase and decrease of the graph of the function f (x) = 4 (x 3)2 + 4 6) Find a quadratic equation through the points (0,0), (,8.5) (5, 2.5)**NO CALC What is the lowest point of the graph of the quadratic function you found?
7) Find the absolute value of the complex number z = 3 + 2i 8) Simplify i 2 + i 40 i 004 + 2i 9) Simplify i ( 2 i ) 2 + 3i(+ i) 0) Simplify 8 4 + 4(+ 6) 2 7 **For graphs- see above CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 5 7 a b 9 ) Simplify 2) Simplify b a 7 27x 9 y 3 ( ) 2 3 (2x 5 y ) 3) Solve for x 27 3 x x 5 = 92x 8 x 2 = 6 4) Simplify log 7 7 + log 4 2 log ( log 7 5) ( log 3 7) ( log 2 27) ( log 5 2) 5) Evaluate 3log 24 24 2 3 log 4 8 log 5 5 log 30 log 9 27
6) Solve for x log x 6 = 2 3 log 9 x = 2 log 4 x = 3 2 7) Expand log 2 ab z 3 2 x log 2 5y 2 z 8) Condense 2log9 + 5logx + log 3 2(ln2 lnx) + (lnx ln4) lny + 2ln(y + 4) 3 [ ] ln(y ) 4 lna ( lnb 2lnc) + lnd 9) Solve Go as far as you can without a calculator. 2 5 2 x 9 = 3 2ln(-x) + 7 = 4 2 e3x = 20
0) You deposit $300 in an account paying 6.5% annual interest compounded continuously. How much is in your account after 6 years? (calc okay) How long will it take your money to triple? ) Loah Nimmer is driving along a straight, level highway at 64 kilometers per hour when his car runs out of gas. As he slows down, his speed decreases exponentially with the number of seconds since he ran out of gas, dropping to 48 km/h after 0 seconds. a) Use exponential regression to write an equation expressing speed in terms of time in the form y = a ib x b) Use your model to predict Loah s speed after 25 seconds. c) At what time will Loah s speed be 0 km/h? d) Draw the graph of the function for speed in the domain from 0 through the time when Loah reaches 0 km/h. e) What would the actual speed-time graph look like for negative values of time?
CHAPTER 8 RADICALS ) Solve for x. Check for extraneous solutions 3 3 2 0 3x = 2 x 8x + = x + 2 x 2/3 +3 =7 x + 5 + x = 5 5 2x 3 = 64 x + 2 + x 3 = 5 2) Simplify. Your answer should contain only positive exponents with no fractional exponents in the denominator. ( x 3 y 2 ) 3 2 x y 2 3 4 a b 3 i a 4 3 b 2 2 3) Evaluate the continued radical 2 + 2 + 2 +... 3 3 3 3... 4) Rationalize the denominator (write in simplest radical form) 4 2 5 3 3 5 ( ) 2 3 5) Simplify 25 5 3 2ab 4 i 3 3 8a 2 b 2 x + y