Algebra Mid-Term STUDY GUIDE A., A., A.4, A.6, A.7, A.9, A.0 Name: Date: Block: The 8 LEARNING TARGETS on the Mid-Term are listed below: Use the order of operations (PEMDAS) to evaluate a numeric expression Use the order of operations to evaluate an algebraic expression using substitution for the variables. Translate verbal expressions in English into algebraic expressions, equations, or inequalities. Solve equations, including those using grouping, fractions, and that have variables on both sides of the equal sign. Know when there are no solutions to equations, or all real numbers are solutions. Know how to justify steps using the properties of equality. Simplify radicals, including simplifying irrational denominators Find the mean, median, mode, and range for a set of data, and compare these measures. Identify the parts of a box-and-whisker plot, and compare data in these plots. Find the Mean Absolute Deviation (MAD), standard deviation (), and variance ( ) for a set of data. Find the z-score for a data point in a set of data. Given a z-score, mean, and standard deviation, find the data point with that z-score. Interpret data based on z-score. Use the formulas to calculate MAD,,, and z-score related information. Plot points on a Cartesian coordinate plane. Find the (x, y) coordinates of points on the Cartesian plane. Identify the quadrant in which a point is located. Given a relation in different formats (such as a set of (x, y) pairs, mapping diagram, or graph), determine whether the relation is a function. Be able to explain why you made your choice. Remember: a relation is a function if every input has one and only one (OAOO) output. Given a relation with a discrete domain, be able to find the range of the relation. Given a graph of a relation, be able to describe the domain and range in set builder notation. Be able to graph linear equations using the table of values method and the intercept method. Be prepared to write the x- and y-intercepts in (x, y) pair form. Be able to evaluate functions that use f(x) function notation. Remember: x is the input to the function, and f(x) is the output.
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page Be able to identify the zero of a linear function, which is the same as its x-intercept. Find it by setting the function s output to 0, and solving for x (find x where f(x) = 0). Be able to put a linear equation into function form (solve for y). Be able to solve formulas for a given variable. Be able to use formulas created by solving for a variable in word problems. Determine the slope of a linear equation Graph linear equations by rewriting equation in slope-intercept form (solve for y), graphing the y-intercept (b), then using the slope to go up or down the change in y (numerator of slope) and left or right the change in x (denominator of slope) to find other points. And finally, connecting points. Identify direct variation equations in the form of y=mx Practice questions: ) Write an algebraic expression for the following statements: a) the difference of three squared a number b) A number less than the quotient of a the same number and nine c) The product of a number and different number d) The sum a number and the square root of 5 e) One-fifth of a number less than negative one f) The quotient of a number cubed and a different number squared g) Five less than four cubed h) The sum of a number and one-third of the same number i) Thirty-one more than one-half of a number j) The product of six and one tenth. ) Write an equation or inequality for the following statements, then solve it. a) Two more than twice a number is six. b) Forty less than five times a number is the same as zero. c) The sum of five and a number is at least four. d) Twice the sum of r and ten is identical to r minus 4. e) Five more than twice a number is three times the difference of that number and two. What is the number?
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page f) Twice the sum of four and a number is six less than that same number. What is the number? g) The quotient of thirty nine and a number is three. What is the number? h) The product of a number and another number is at most ten (don t solve this one). i) The quotient of two different numbers is no more than sixteen (don t solve this one). j) Three times a number decreased by four is at least five times a different number (don t solve this one). ) Use the order of operations to evaluate the expressions: a) 6 + 4 b) 6( - 7) - 4( + ) c) (-) + 6( 5) 9 d) e) 8 [ + (6 + )] f) - + 6-5 g) 58 + 6 6 - h) (4 (-) + (-) ) 7 i) 6 ( + ) j) 75 [6(8-6) + ] k) 6 + 4[ + ( - 8) ] 4) Substitute the given variable values in the expressions and evaluate: a) (x y) (y + z) for x =, y = -4, z = -5 b) x x + y for x = -, y = c) + x + x when x = -4 d) x +5(y ) for x=-, y=5 e) ( y x) (z x) + y z for x=, y=8, z=9 5) The following problems require two steps.
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 4 First, translate the English phrase into a math expression using variables. Second, substitute the variables in your expression with the values x = - and y = -4. a) Five less than three times a number x. Translation: Value: b) Six more than the quotient of a number y squared and x. Translation: Value: 6) A skating rink holds parties and charges a rink rental fee of $00, plus $0 per person. a) Write an expression that the rink can use to determine how much to charge. b) Use the expression you wrote in part a) to determine how much will be charged if 50 people attend a party. c) How much would it cost if 50 people attend and the customer has a coupon for $00 off the total? 7) The area of a triangle is given by A = bh where b = the length of the base and h = the height of the triangle. If the height of a triangle is times its base, what is its area if its base is 8 inches? a) Write an expression that describes the situation in terms of the base, and b) find the value of the area of the triangle. 8) The first stage of a rocket burns 8 seconds longer than the second stage. If the total burning time for both stages is 5 seconds, how long does each stage burn? 9) The length of a rectangular map is 5 inches and the perimeter is 50 inches. Find the width. ( perimeter = L + W where L is the length and W is the width) 0) Karin s mom runs a dairy farm. Last year Elsie the cow gave 75 gallons less than twice the amount from Bessie the cow. Together, Elsie and Bessie produced 464 gallons of milk. How many gallons did each cow give? ) Janice is considering purchasing a refrigerator from three different dealers. Dealer A has given a good price. Dealer B's offer was $00 more than Dealer A's price. Dealer C said he'd double Dealer A's price, but then subtract $500. If Dealer B and Dealer C had the same price, what was Dealer A's offer? ) In 00, Stock A and Stock B had the same price. Since then, the price of Stock A has tripled. In the same time span, the price of Stock B has increased by $500. If both stocks are once again at the same price, what was the original price? What is the price of each now?
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 5 ) Estimate to the nearest whole number: a) 40 b) 0 Simplify the radicals: 4) 50 5) 48 6) 4 7) 6 8) 5 9) - 7 6 0) 54x 4 y ) x ) 0x y 4 z 7 ) 9x 4x Rationalize the denominators 4) 5) 6) 7) 5 00 5 8 x 5 x y 4x 8) 9) Solve the equations, justifying steps for the first 5 problems. Verify your solutions for all problems by substituting the solution back into the equation. a) x = 4 b) 4 x = 4 c) x + = 4
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 6 d) 9x 4x 5 = 0 e) 5x 4(x-) = 7 f) 0 + x = 4x 7 g) 0z = 5(z ) h) 8(x + ) - = 8x + i) 4x (x + ) = -5 j) x + 6 + x = 4x 6 x k) 0x - (4 + 5x) = 8 l) 5x 7 = x - m) 7 = x x + 5 n) step : 6(x - ) = 9 x step : x - 6 = 9 x ) What field property justifies step? ) Finish solving equation. 0) Solve the inequalities and graph the solutions on a number line. a) 4(x ) < -x +5 b) -4x < - c) x + 7 Sample questions (round to the nearest hundredth as necessary): ) Judy likes to track her savings account. The box and whisker plot shows how much money she had daily over the past two months. a) What is the range of the amount of money in Judy s account? b) What is the inter-quartile range of the amount of money in Judy s account? c) What fraction of the time was the account between $5and $68? d) What percent of the time is the account between 8 and 87? e) What is the value of the lower quartile? f) What is the value of the lower extreme? g) What is the value of the median? h) What fraction of the time did the account have $5 or less? i) Which quarter has the data the most spread out? j) Which quarter has the data the most clustered? k) What percent of the data falls between the upper quartile and the upper extreme?
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 7 l) What percent of the data is greater than the lower quartile? ) Part : The chart at right shows the scores for each of Sam and Mary s math grades for a school term. Using this chart, create a stacked box and whisker plot to compare Sam s and Mary s math grades. ) Part : Using the data from the same chart at right, which of the following is true? a) The range for Sam s dataa is less than Mary s range. b) The mean for Mary s dataa is the same as the median for Sam s data. c) Mary s mean score is less than Sam s mean score. d) Mary s median score is higher than her mean score. The stacked box and whisker plots above track the stock performance of four leading companies. a) Which company holds the record for the most stocks sold? How many did they sell? b) Which company has the greatest median amount of stocks sold?
4) Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 8 The box-and-whisker plot above compares homework time and tv time for a class of 8 th grade Algebra students. Which statement is false? a) The upper quartile for homework time is the same as the median for tv time. b) The interquartile range for tv time is less than that for homework time. c) The range for tv time is greater than the range for homework time. d) The MAD for homework time is most likely less than the MAD for tv time; 5) Find the mean, mean absolute deviation (MAD), variance ( ), and standard deviation () for the data sets below: a) {, 8, 5, 0, 50} b) {96, 8, 00, 90, 7, 8} c) {450, 600, 4, 8, 550, 50, 475 6) Suppose the data from question 5c above represents the Algebra SOL scores for some students. a) Find the z-score of each score b) How many of these SOL scores fall within one standard deviation of the mean? c) How many standard deviations are required to contain all of the data? 7) Alice and Martin are in the for this test: same French class. The following represent class scores {00, 8, 97, 75, 85, 8, 85, 90, 9, 70, 66, 80} a) Find the mean and standard deviation for the data. b) Alice s test score was a 97. How many standard deviations is her score from the mean? c) Martin s scored 0.89 standard deviations below the mean. What was his test score (to the nearest whole number)?
8) Barbara took two tests: Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 9 On her English test, she received a score of 8. The mean of the class was a 5 and the standard deviation was 0.5. On her physics test, she received a score of 5. The mean of the class was 49 and the standard deviation was 0.8. On which test did she do better? Explain your answer. 9) The mean height for US males is 70 inches with a standard deviation of.8. The mean length of a cat is 8 inches with a standard deviation of.. Mr. Jones is 66 inches tall, and his cat Mr. Tinkles is 6 inches long. Who is shorter, compared to average height for their respective species? Explain your answer. 40) The data set shown has a mean of 65.4 and a standard deviation of 6.7, rounded to the nearest tenth. {, 6, 4, 4, 47, 5, 6, 75, 87, 98, 0, 0} How many of these data points have a z-score greater than -0.4? 4) A data set has a mean of 5.4 and a standard deviation of 6.5. An element in this data set is 4. What is the z-score for 4? Round to the nearest hundredth. 4) A data set has a mean of 4.7 and a standard deviation of 8.9. An element in this data set is. standard deviations above the mean. What is the value of the element?
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 0 4) The table shows the scores of an Algebra test for four different classes. It also shows the z-score of one student in the class. Which student had the highest score on the test? Mean for the class Standard Deviation for the class Student s z-score Bob 8 0. Morris 88 9. Kathy 84.4 Robin 79 6.8 44) From 984 to 995, the winning scores for a golf tournament were: 76, 79, 77, 78, 78, 80, 8, 79, 85, 7, 79, and 78 a) Find the mean and standard deviation for the data (round to the nearest hundredth). b) Find the z-score of the first 4 unique scores in the list (76, 79, 77, 78) c) How many of these winning scores fall within one standard deviation of the mean? 45) A math class took a test, the results yielded a mean of 89 with a standard deviation of 4. If students have a z-score less than -, they have to do a re-take. Use the table below to determine the z-scores for each student and which of these students have to do a retake. Name Statistics Test Score Andrea 77 Nora 8 Rosella 84 Dona 80 Nikole 8 Emily 9 Z-Score Retake Yes/No
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 46) Use the graph at right to answer the questions a) Plot and label the points P( (-, -), Q(, 0), R(0, ), and S(4, -5) in the coordinate plane at right. Identify the quadrants of the points. b) Identify the coordinates of points A, B, and C. Identify the quadrants they lie in. 47) Determine whether the relations below are functions or not. Explain how you made your choice. a) {(-4, ), (0, -4), (4, ), (, -4)} b) c) d) Function? Explain: Function? Explain: Function? Explain: Function? Explain: 48) Given the functions below, find the range given their domain. a) y = -x + D= {-, -, 0,, 4} b) y = x - D= {-4, -, 0,, 6} 49) Given the graphs below, determine whether the relations are functions. Then find the domain and range for each in set builder notation.
a) Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page b) c) d) Function? Domain Range Function? Domain Range Function? Domain Range Function? Domain Range 50) Graph the function with the given domain using the table of values method. Then identify the range of the function in set builder notation. The domain is the set of all real numbers, D = {x x}. a) y = 4x + b) y = - x - 5) Find the x- and y-intercepts of the graphs of the equations as ordered pairs. Then graph the functions using the intercept method.
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page a) x y = b) 4x y = -8 c) y = - 4 x + x-intercept y-intercept x-intercept y-intercept x-intercept y-intercept 5) Evaluate the functions below. a) f(x) = -5x + 4 Find: f(), f(-), f(0) b) g(x) = x 7 4 Find: g(0), g(4), g(-8) c) h(x) = x + 9 Find: h, h(0), h(9) Find the zero of h (the x-intercept) 5) Given f(x) = 4x, complete the table below for the function. X - 0 f(x) 0 7 54) Find the zero of each function (the x-intercept) below. a) f(x) = 4x + 6 b) h(x) = x 0 55) Put the linear equations below into function form (solve for y). a) 5x + 0 = 5y b) -x + 4y = 4 x 6y c) = d) 0x y + 0 = 0 56) Solve each equation for the given variable.
a) Solve for t: v = r + at Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 4 b) Solve for x: c) Solve for a: 5xy + n 4a + b = a = 6 d) Solve for n: n s = (a + t) e) Solve for c: x + y = 4 c f) Solve for C: F = 9 C+ 5 57) The area of a trapezoid is given by A = h(b + b), where b and b are the lengths of the bases of the trapezoid. a. Solve the formula for h, the height of the trapezoid. b. Use the formula you created in part a above to find the height of a trapezoid if its area is 40 square inches, and its bases are 0 and 6 inches in length. 58) Find the slope of the line that passes through the points. a) (, -) and (-, ) b) (-, -) and (4, -) c) (, 5) and (, -6) d) (0, ) and (-, 5) e) (-, 4) and (-5, 8) f) (-, ) and (-6, 0) 59) Could any of the lines described in the previous problem be parallel? Explain your reasoning. 60) Find the slopes and y-intercepts of the lines below: a) y = -x - 6 b) x + 8y = 6 c) x 4y = 6 6) Graph the functions using the slope-intercept method. Remember to put the equation into slope-intercept form if necessary. Identify the slope and the y- intercept.
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 5 a) y = x 5 b) y = - x slope y-intercept slope y-intercept c) y x = 4 slope y-intercept d) 4x + y = slope y-intercept 6) Identify the slope of the lines shown: a) b) c) 6) At the beginning of the day, you had 0 gallons of gas in the car. After driving for 4 hours, you had gallons of gas in the car. What was the rate of change? 64) Which equations below show direct variation? If an equation does show direct variation, what is the constant of variation? a) y = - x b) x 4y = 0 c) 9x + 0y = d) -x y = 0 65) Given the y varies directly with x, write a direct variation equation that relates x and y. a) x = -0, y = 5 b) x =, y = c) x = -, y = d) x = 6, y =- 66) Which of the graphs below show direct variation? There may be more than one.
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 6 a) b) c) d) 67) Which of the tables below show direct variation? If so, what is the direct variation equation. There may be more than one table that shows direct variation. a) b) 68) One variable (A) varies directly as the other (C). Find the missing numbers x and y. Write the formula which relates the variables. 69) There are about 00 calories in 50 grams of Swiss cheese. Willie ate 70 grams of this cheese. About how many calories were in the cheese that he ate if the number of calories varies directly as the weight of the cheese. 70) The resistance (R) of a copper wire varies directly as its length (L). Write this relation as a direct variation using k as the constant of variation. 7) The distance an object, a, drops from rest in freefall varies directly with the square of the time, t. If a varies directly as t, and a = when t =, find a when t =. 7) Graph the equations. What are the slopes of the lines? a) y = 4 slope b) x = - slope
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 7 ANSWERS x ) a) b + 4 b) 0 c) x xy + 8 d) 9n n e) m + 5 6 8 4 f) x g) 5x 5 h) x + x 4 i) 50 y x j) x ) a) 5x 4 = 6 (x = ) b) 5+4q (q -8) c) 5+x 4 (x-) d) (r + 0) = r 4 (r=-4) e) x+5=(x-) (x=) f) (4 + x) = x - 6 (x= =-4) 9 g) = (x=) h) xy 0 x x i) 6 j) x - 4 5y y ) a) b) c) - d) e) 4 f) 4 g) 7 h) 9 i) 9 j) 60 k) 7 7 4) a) b) -6 c) 7 d) 6 e) 6 5 y 5) a) Translation: x 5 value: - b) Translation: + 6 value: - x 6) a) y = 00+0x b) $600 c) $400 7) a) A = (b)(b) = ()b = b b) A = 64 in 8) x + 8 + x = 5; first stage (x +8) = 90; second stage (x) = 6 9) (5)+w = 50; w = 0 0) E=B 75; E+B = 464; B-75+B=464; Bessie = 6 gals; Elsie = 85 gals ) A+00 = A 500; A = $6000 ) A=B; A = B+500; A = A+ +500; A = B = $750 ) a) 6 b) 4) 5 0 5) 4 9) - 0) xy 4) 0 5) 4 x ) x ) 4xy z 5xz 9) a) x = 6 (addition prop. of equality) b) x = 8 (add. prop of =; mult. prop of equality) c) x = 4 (subtraction prop. of equality; division prop of equality) d) x = 5 (combine like terms; add. prop. of equality; div. prop. of equality) e) x = 5 (distributive prop.; combine like terms; subtraction prop. of equality) f) x = 9 g) no solution h) all real numbers i) x = j) no solution k) no solution l) x = m) x = n) ) distributive property ) x = 0) a) x < b) x > c) x 6) 6 7) 6 6 8) 5 6) x x 5 xy 7) y ) 6x x 8)
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 8 Statistics Test Study Guide Answers ) a) 8 b) 5 c) 4 d) 75% e) 5 f) 8 g) 68 h) 4 i) first j)fourth k) 5% l) 75% ) Part : Part : b ) a) company D; about 450 b) company C 4) b 5a) = 5, MAD = 4, =.6, = 5.8 5b) = 87, MAD = 8., = 9.67, = 9.6 5c) = 484, MAD = 6.9, = 554, = 7.79 6a) z-scores: -0.47,.6, -.0, -.4, 0.9, 0.50, -0. 6b) 4 scores (450, 550, 50, 575) 6c) standard deviations are needed to contain all data 7a) =8.75, = 9.87 7b).4 standard deviations above the mean 7c) Martin received a 75 on the test. 8) Barbara s z-score for her English test is 6, and her z-score for her physics test is.75. She did better on her English test, because she was 6 standard deviations above the mean, while she was only.75 standard deviations above the mean for the physics test. 9) Mr. Tinkles is comparatively shorter than Mr. Jones because Mr. Tinkles z-score is -.67 while Mr. Jones z-score is approximately -.4. So Mr. Tinkles is more short (.67 inches under the mean) than Mr. Jones (.4 under the mean) compared to their species. 40) 6 points (all the points greater than 54.7) 4) -.79 4) 6.09 4) Kathy (got a 99.4) 44) a) µ=78.58; =.0 b) -0.86, 0.4, -0.5, -0.9 c) 9 points (between approximately 75 and 8) 45) Name Statistics Test Score Z-Score Retake Yes/No Andrea 77 - Yes Nora 8 -.5 No Rosella 84 -.5 No Donna 80 -.5 Yes Nikole 8 - No Emily 9.5 No Andrea and Donna have to re-retake the test.
46) a) P III, Q no quadrant, R no quadrant, S IV b) A(-, -4) III, B(0, -4) no quadrant, C(-, 4) II Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 9 Study Guide Answers 47) a) Yes, every input has OAOO output. b) No, input 5 has more than one output. c) Yes, every input has OAOO output. d) No, there exist inputs with more than one output. 48) a) R ={, 8,, -, -0} 5 b) R = {-5, -4, -,, 0} 50) (Tables may vary depending on inputs b) chosen) a) 49) a) No; D={x R x y} b) Yes; D = {x x}; R = 0}; R = {y x R {y y} c) No; D = {x -5 x < 0}; R = {y - < y < 4} d) Yes; D = {x -4 x 4}; R = {y - y 4} 5) a) x-int: (4, 0); y-int: (0, -6) b) x-int: (-, 0); y-int: (0, 8) c) x-int: (4, 0); y-int: (0, ) 5) a) f()=-, f(-)=9, f(0)=4 b) g(0)=-7, g(4)=-4, f(-8)=- c) h =8, h(0)=9, h(9)=6 zero of h occurs at x=- 5) 54) a) x=-4 b) x=5 55) a) y = x + b) y = x + 6 c) 4 d) y = 5x + 0 A 57) a) h = b) 5 inches b + b v r --n y = x 4 56) a) t = b) x = c) a = -b a 5y s x + y 5 d) n = e) c = f) C = (F ) a + t 4 9
Algebra Mid-Term STUDY GUIDE A., A., A.4, A.7, A.9, A.0 Page 0 Study Guide Answers 4 4 66) a) and d) are direct variations 58) a) b) 0 c) undefined d) e) - f) 59) The lines in a) and d) could be parallel 67) a) not direct variation b) yes, N = M because they have the same slope. 68) y=6, x = 5 equation: C = A 60) a) m=-, b=-6 b) m =, b= c) m=, b=- 4 6) a) slope =, y-int =-5 4 69) 80 calories 70) R = kl b) slope = -, y-int = 0 7) a = 7 when t = c) slope =, y-int = 4 (rewrite as y = x+4) 7a) m = 0 d) slope = -, y-int=6 (rewrite as y=- x+6) b) undefined slope 6) a) m=- b) y=- c) x = 6) - gal/hour (take slope of points (0, 0) and (4, )) 64) a) yes; k = b) yes; k= c) no d) yes; k=- 65) a) y=- x b) y = 6x c) y = -4x d) y = - x