Survey
* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project
Download Here - Mr. Young Math
Transcript
Name ______________________________ Teacher ____________________________ Page | 1 Algebra I Per: ______ Page | 2 Unit 1: Statistics Topic A: Distributions and their Shapes Dot Plots: Each dot represents one piece of data in the set. The dot corresponds to the value that the dot is placed over on the number line. Histograms: A graph of data that groups the data based on intervals and represents the data in each interval by a bar. Shows the general shape of a distribution Box Plots: A graph that provides a picture of the data ordered and divided into four intervals that each contains approximately 25% of the data. Mean: Average (add all of the data values and divide by the number of values) Balance point: the sum of the distances to the right of the mean is equal to the sum of the distances to the left of the mean. Median: Middle value in a list of data in numerical order If there is an even number of values, the median is the average between the two middle values Mode: Data value that occurs the most times There can be more than one mode or no mode Range: Max Value – Min Value Practice Problems: 1. Sam said that a typical flight delay for the sixty BigAir flights was approximately one hour. Do you agree? Why or why not? Page | 3 2. Here are the data for ten students who were asked how many hours they spent watching TV last night. a. Construct a dot plot of the data on hours of TV. Would you describe this data distribution as approximately symmetric or as skewed? b. If you wanted to describe a typical number of hours of TV for these ten students, would you use the mean or the median? Explain and calculate the values of each. Student A B C D E F G H I J Hours of TV 2 1 0 3 4 1 2 2 4 3 Types of Distributions: Symmetrical: Mean and Median are approximately equal Usually use Mean as an approximation of the data Skewed: Median is the best approximation of the data Mound Shaped Uniform Skewed with a tail to the right Mean will be pulled by the “tail” Skewed with a tail to the left Page | 4 U Shaped Mean and Median are approximately equal but neither is a good indicator of typical data. Describing Variability and Comparing Distributions Deviations from the mean= Data value minus the mean (algebraically: x x ) The greater the variability (spread) of the distribution, the greater the deviations from the mean (ignore deviation signs) High Variability of data (wide spread) Large Deviations Dotplot of Value High Standard Deviation Low Variability of data (small spread) Small Deviations Low Standard Deviation Less predictable More predictable 3. Look at the dot plot below. 0 1 2 3 4 5 Value 6 7 8 9 10 a. Calculate the mean of this data set. 4. Three data sets are shown in the dot plots below. Data Set 1 Data Set 2 Data Set 3 20 21 22 23 24 25 26 27 28 29 30 a. Which data set has the smallest standard deviation of the three? Justify your answer. b. Which data set has the largest standard deviation of the three? Justify your answer. Page | 5 To Calculate Standard Deviation 5. Use your calculator to find the sample standard deviation (Sx), to the nearest tenth, of a data set of the miles per gallon from a sample of five cars. 24.9 24.7 24.7 23.4 27.9 6. Suppose that a teacher plans to give four students a quiz. The minimum possible score on the quiz is 0, and the maximum possible score is 10. a. What is the smallest possible standard deviation of the students’ scores? Give an example of a possible set of four student scores that would have this standard deviation. Draw a dot plot. b. What is the set of four student scores that would make the standard deviation as large as it could possibly be? Use your calculator to find this largest possible standard deviation. Draw a dot plot. Parts of a Box Plot: Minimum: Smallest value in the data set Maximum: Largest value in the data set Median: Middle of Data set (50% of data is above, 50% below) Q1: Lower Quartile: Median of lower half of data (25% of data is below Q1) Q3: Upper Quartile: Median of upper half of data (75% of data is below Q3) Each piece of box plot contains 25% of the data Mid- 50 is data from Q1 to Q3 (50% of data) Interquartile Range (IQR): Q3 – Q1 Describes variability of skewed distribution (High IQR = high variability) Outliers are values more than 1.5(IQR) from the nearest quartile Page | 6 7. A data set consisting of the number of hours each of 40 students watched television over the weekend has a minimum value of 3 hours, a Q1 value of 5 hours, a median value of 6 hours, a Q3 value of 9 hours, and a maximum value of 12 hours. a. Draw a box plot representing this data distribution. b. What is the interquartile range (IQR) for this distribution? c. What percent of the students fall within this interval? 8. Using the histograms of the population distributions of the United States and Kenya in 2010, approximately what percent of the people in the United States were between 15 and 50 years old? Approximately what percent of the people in Kenya were between 15 and 50 years old? a. What 5-year interval of ages represented in the 2010 histogram of the United States age distribution has the most people? b. Why is the mean age greater than the median age for people in Kenya? Page | 7 Categorical Data on Two Variables Categorical data are data that take on values that are categories rather than numbers (such as gender) A two-way frequency table is used to summarize bivariate categorical data. Super Telepathy To Fly Freeze time Invisibility Strength Females (a) (b) (c) (d) (e) Total (f) Males Total (l) (r) (g) (m) (h) (n) (i) (o) Joint frequency: a, b, c, d, e, g, h, I, j, k Marginal frequency: f, l, m, n, o, p, q (j) (p) (k) (q) Relative Frequency: compares a frequency count to the total number of observations (Divide frequency by total and write as a decimal or percent) Super Telepath To Fly Freeze time Invisibility Total Strength y Females a/r b/r c/r d/r e/r f/r Males g/r h/r i/r j/r k/r l/r Total m/r n/r o/r p/r q/r r/r The marginal cells in a two-way relative frequency table are called the marginal relative frequencies, while the joint cells are called the joint relative frequencies. Conditional Relative Frequency compares a frequency count to the marginal total that represents the condition of interest. Divide frequency by the total of the condition of interest Condition of interest: Gender To Fly Freeze time Invisibility Super Strength Telepath y Total Females a/f b/f c/f d/f e/f f/f Males g/l h/l i/l j/l k/l l/l The differences in conditional relative frequencies are used to assess whether or not there is an association between two categorical variables. The greater the differences in the conditional relative frequencies, the stronger the evidence that an association exits. Page | 8 9. Juniors and seniors were asked if they plan to attend college immediately after graduation, seek full-time employment, or choose some other option. A random sample of 100 students was selected from those who completed the survey. Scott started to calculate the relative frequencies to the nearest thousandth. Plan to attend Plan to seek fullOther options Totals College time employment Seniors Juniors Totals a. Complete the calculations of the relative frequencies for each of the blank cells. Round your answers to the nearest thousandth. b. A school website article indicated that “A Vast Majority of Students from our School Plan to Attend College.” Do you agree or disagree with that article? Explain why agree or why you disagree. 10. Scott started to calculate the row conditional relative frequencies with grade as the condition of interest to the nearest thousandth. Plan to Attend College Plan to seek FullTime Employment Other Options Totals Seniors Juniors a. Complete the calculations of the row conditional relative frequencies. Round your answers to the nearest thousandth. b. Are the row conditional relative frequencies for juniors and seniors similar, or are they very different? c. Do you think there is a possible association between grade level (junior or senior) and after high school plan? Explain your answer. Page | 9 Numerical Data on Two Variables A scatter plot can be used to investigate whether or not there is a relationship between two numerical variables. Types of relationships Linear Quadratic Exponential Least Squares Line (line of best fit) is used to model a linear relationship: y = ax + b ~provides prediction of a y value for a given x value To find the equation in the calculator: Step 1: press STAT. Step 2: From the STAT menu, select the EDIT option. (EDIT enter) Step 3: Enter the x-values of the data set in L1. (to clear lists press the up arrow to highlight L1 or L2 and press CLEAR not delete) Step 4: Enter the y-values of the data set in L2. Step 5: Select STAT. Move cursor right to the menu item CALC and then move the cursor to option 4: LinReg(𝑎𝑥 + 𝑏); Press enter. Step 6: Put in L1 (2nd #1) comma L2(2nd #2) and press enter. Page | 10 11. Below is a scatter plot of foal birth weight and mare’s weight. 130 Foal Weight (kg) 120 110 100 90 0 a. 0 500 510 520 530 540 550 560 Mare Weight (kg) 570 580 590 The equation of the least squares line for the data is: (in kg) and 520 kg? , where = mare’s weight = foal’s birth weight (in kg). What foal birth weight would you predict for a mare who weighs b. How would you interpret the value of the slope in the least-squares line? c. Does it make sense to interpret the value of the -intercept in this context? Explain why or why not. Prediction Error: Residual = actual y-value – predicted y-value Residuals are the vertical distances of the points from the least-squares line. Actual y value = data value from table Predicted y value = y value from plugging corresponding x value into the least squares line a. Meerkats have a gestation time of 70 days. Use the equation of the least-squares line , (where x = number of days) to predict the longevity (y) of a meerkat. b. The longevity of the meerkat is actually 10 years. Use this value and the predicted value that you calculated in (a) to find the residual for the meerkat. Page | 11 To make a residual plot, plot the x-values on the horizontal axis and the residuals on the vertical axis. Correlation Coefficient (r): measures strength of the relationship close to 1 or -1 = strong; close to 0 = weak If r is positive then there is a positive slope to the least squares line; if r is negative there is a negative slope to the least squares line Strong Linear Relationship Residual Plot will have NO pattern (random scatter) Weak or no linear Relationship Residual plot might show a quadratic or exponential (curved) relationship Correlation Coefficient will be close to 1 or -1 Correlation Coefficient will be close to 0 12. If you see a random scatter of points in the residual plot, what does this say about the original data set? 13. Suppose a scatter plot of bivariate numerical data shows a linear pattern. Describe what you think the residual plot would look like. Explain why you think this. 14. If you see a clear curve in the residual plot, what does this say about the original data set? 15. If you see a random scatter of points in the residual plot, what does this say about the original data set? Page | 12 16. The scatter plot below displays data on the number of defects per 100 cars and a measure of customer satisfaction (on a scale from 1 to 1000, with higher scores indicating greater satisfaction) for the 33 brands of cars sold in the United States in 2009. Data Source: USA Today, June 16, 2010 and July 17, 2010 a. Which of the following is the value of the correlation coefficient for this data set: , , , or ? b. Explain why you selected this value. NEW MATERIAL! 17. Joshua scored a 65, 80, 85, and 100 on his first four math quizzes. What score will Joshua have to earn on his fifth quiz to have an average score of 85? 18. Below are the salaries (in thousands of dollars) of Varsity coaches at Smithtown West. If the mean salary is $5,000, find the missing value in the table Page | 13 Unit 2: Foundations of Algebra Types of Functions Linear – y = mx + b m = slope, b = y-intercept Exponential – y = ax where 0 < a < 1 (a > 1 growth, a < 1 decay) Quadratic – y = ax2 + bx + c where a 0. Practice Problems 19. The graph below shows the total cost and the revenue a company makes manufacturing flash drives for Staples. a. Write an equation for each line in the graph b. Why does the cost equation start at $4000? c. How many flash drives does the company have to make and sell in order to make a profit? 20. Every morning Tom walks along a straight road from his home to a bus stop, a distance of 160 meters. The graph below shows his journey on one particular day. a) Make a distance from home vs time story for this graph b) Over which time interval is the rate of change 4 meters per second? c) What happens at 100 seconds? Page | 14 21. The bee population doubles every hour during early spring. If 10 bees are in the hive at the start, graph the population of bees for the next 6 hours. Label and scale the y-axis appropriately. a) Show a table of values here for your plots b) How many bees are in the hive after 4 hours? __________ c) What type of graph is this? _________________________ 22. Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players remain after 5 rounds? 23. The graph of a function is shown below. a) Over which of the following intervals on the x axis is the function decreasing (has a negative rate of change) ? (1) x = 0 to x = 2 (2) x = -2 to x = 0 (3) x = -6 to x = -2 (4) x = 1 to x = 3 b) What type of function is this? Page | 15 24. What type of function would best model the elevation vs time of a ball thrown straight up into the air? Why? 25. A plane flying from San Francisco to Denver is modeled by the linear function A(t) = 33,000 – 120t where t represents time and A(t)represents altitude. i. What is the slope? What does the slope mean in context to the question? ii. What is the y-intercept? What does the y-intercept mean in context to the question? 26. A new computer repair company offers house calls to their customers. They use the function f(h) = 45h + 80 to calculate the total repair bill, f(h), for h number of hours worked. i. What is the slope? What does the slope mean in context to the question? ii. What is the y-intercept? What does the y-intercept mean in context to the question? iii. How much would the repair bill be if a worker spent three hours on a house call? Properties of Real Numbers / Operations with Polynomials Property Example Commutative Property of Addition or Multiplication (order doesn’t affect straight addition or multiplication) 1 3 31 Associative Property of Addition or Multiplication (grouping doesn’t affect straight addition or multiplication) (1 2) 3 1 ( 2 3) Distributive Property (Multiplication can be “distributed” over addition or subtraction) 2(5 3) 2(5) 2(3) 2(3) 3( 2) 2(3 4) ( 2 3)4 2(5 3) 2(5) 2(3) Page | 16 ADDING/SUBTRACTING POLYNOMIALS: Combine the coefficients of like terms. (EXPONENTS STAY THE SAME!) MULTIPLYING POLYNOMIALS: Multiply coefficients, add exponents of common variables. Example: What is the product of -3x2y and 5xy2 + xy? -3x2y(5xy2 + xy) = -15x3y3 – 3x3y2 Multiplying two Binomials: Double Distribute or FOIL (First, Outer, Inner, Last) (c + 8)(c – 5) = c2 – 5c + 8c – 40 = c2 +3c – 40 (x – 2)2 = (x – 2) (x – 2) = x2 – 2x – 2x + 4 = x2 – 4x + 4 27. While solving the equation Which property did she use? , Becca wrote 1) 2) 3) 4) distributive associative commutative identity 28. Make a flow diagram to show that a+(b+c )=c+(b+a) . 29. Use the abbreviations “C” for Commutative Property and “A” for Associative Property to complete the flow diagram Page | 17 30. Find each sum or difference by combining the parts that are alike a. (3x2 – 4x + 2) + (x2 + 3x – 7) b. (5x2 + 3xy – 4) – (x2 + 4xy – 7) c. From (5x2 – 12x + 8) subtract (3x2 – 2x + 4) d. Subtract (6x2 – 3x – 1) from (-2x2 + 4x – 5) 31. Describe the property or operation used to convert the equation from one line to the next: 8y – (8 + 6y) = 20 8y – 8 – 6y = 20 ____________________________________________________ 8y – 6y – 8 = 20 ____________________________________________________ 2y – 8 = 20 ____________________________________________________ 2y = 28 ____________________________________________________ y = 14 ____________________________________________________ 32. Use the distributive property to write each of the following expressions as the sum of monomials. i. (x + 1)2 b. (x2 + 4)(x – 1) c. (x2 – x + 1)(x – 1) d. (-2x3 – 2x + 1)(x2 – x + 2) Page | 18 33. Draw a picture to represent the expression (a + 6)(2a2 – 3a + 1), and write the product in simplest form. 34. Express as a trinomial in simplest form: (3x – 4)2 Solving Different Types of Equations For an equation to have a solution set of all real numbers one side must be equal to the other through the use of properties of real numbers or operations UNDEFINED FRACTIONS: An algebraic fraction is undefined (not permissible) when the denominator is equal to zero Example: x2 x4 The expression is undefined for x = 4 which is the value that not permissible for x. SOLVING FRACTIONAL/RATIONAL EQUATIONS: o First identify the values that are not permissible (undefined). We call these values restrictions. o Cross multiply, set equal, and solve. Example: Solve for x 2 5 3 x 6 x 2 5 and 3 x 6 x 2(6 x) 5(3 x) x 3,6 12 2 x 15 5 x 3 7x x 3 7 and x 3,6 Literal Equations – An equation where the solutions are non-numerical Example: Solve the equation A p prt A p prt A p t pr Page | 19 Systems of Equations – To solve a system of equations, you can do it algebraically or graphically Graphically – Get each equation in y = mx + b form. State the m and b and graph both. Label the point of intersection (if there is one) and state its coordinates. Algebraically – Elimination or Substitution By Substitution – Solve: x=y–4 7x + 5y = -28 7(y 4) 5y 28 7y 28 5y 28 x (0) 4 12y 28 28 x 4 (x , y) (4,0) 12y 0 y0 By Elimination (Addition) – Solve: 3x – y = 3 x + 3y = 11 1(3 x y 3) 3x y 3 3(x 3y 11) 3 x (3) 3 3x y 3 3x 9 y 33 10y 30 y 3 3x 6 (2,3) x 2 Four Steps to Solve Word Problems – 1) Write Let Statements 2) Write Equation 3) Solve Equation 4) Sub back into Let Statements Things to remember about INEQUALITIES If you multiply or divide by a negative number, you need to switch the sign of the inequality. Closed circles when , Open circle when <, > Compound inequalities have the word AND (BARBELL) or OR (ROWBOAT) Ex. Solve for x, graph your solution set: 1 2 x 5 3 1 2x 5 and 6 2x 3 x ( x 3) and 2x 5 3 2 x 2 x 1 Page | 20 When graphing an inequality in two variables, x and y, solve the inequality for y, then graph it as if it were a line ( or means solid line or means dotted line). When graphing a system of inequalities, just graph the two separate inequalities and label the overlap in the shading with an “S.” 35. Solve for x: a) 2(4x + 8) = 3(3x – 6) b) 4x 3 7x 1 3 4 c) 2(x – 4) + x = 4(x + 6) – 1 36. Consider the rational expression 7x . What value(s) make the fraction undefined? ( x 4)( x 2) Page | 21 37. Given the equation: 2x 3 2 x4 3 A. Determine all values that represent a restriction. B. Solve the equation for x. 38. Solve for y: 3x – 2y + 6 = a 39. Solve for x: a) 2 xy 10a 3 b) ax + b = d – cx c) xb c 4 Page | 22 40. Which of the following equations has a solution of all real numbers? i) 3(x + 1) = 3x + 1 ii) x + 2 = 2 + x iii) 4x(x + 1) = 4x + 4x2 (1) ii only (3) ii & iii (2) i & ii (4) i, ii, & iii 41. Find the solution set to each inequality. Express the solution graphically on the number line: a. 2x < 10 b. -2f < -16 c. 6 – a ≥ 15 d. -3(2x + 4) > 0 42. Solve for x, graph and express your answer in set notation: a) 10 < 3x + 4 or -2x - 5 > 1 b) 5 3 2x 11 43. Multiple Choice: Which ordered pair is a solution to the inequality y > 2x + 5 a) (0, 5) b) (-2 , 3) c) (10, 1) d) (1 , 7) Page | 23 44. Multiple Choice: Which is a solution of 2x + 5 > 1 ? a) -6 b) -5 c) -4 d) -3 e) -2 45. Graph the solution set to this system of inequalities. 1 y x 1 3 x y2 46. Solve the following systems of equations graphically y = -3x – 4 y = 2x + 1 47. What is the value of y in the following system of equations? 2x + 3y = 6 2x + y = -2 Page | 24 48. Solve the following system of equations: x = 3y + 8 4x + 5y = -2 49. Solve the following system of equations algebraically: y = -2x + 7 3x + 2y = 13 50. Solve the following system of equations algebraically: 5x + y = 13 4x – 3y = 18 Creating Equations to Solve Problems Consecutive Integer Problems Let Statements Consecutive Integers: Let x = 1st consecutive integer 28 x + 1 = 2nd consecutive integer x + 2 = 3rd consecutive integer Consecutive Even/Odd Integers: Let x = 1st consecutive even/odd integer x + 2 = 2nd consecutive even/odd integer x + 4 = 3rd consecutive even/odd integer Example: Find two consecutive odd integers whose sum is Let x = 1st consecutive odd integer x + 2 = 2nd consecutive odd integer x + x + 2 = 28 2x + 2 = 28 2x = 26 x = 13 Therefore, the odd integers are 13 and 15. Page | 25 51. Tickets for admission to a high school football game cost $3 for students and $5 for adults. During one game, $2995 was collected from the sale of 729 tickets. How many tickets were sold to the students and how many were sold to the adults? 52. At the movie theatre, Matt bought one soda and two boxes of candy for $5.50. Mike bought three sodas and three boxes of candy for $11.25. Find the cost of one soda and the cost of one box of candy. 53. Find three consecutive integers such that their sum is 51. 54. Find two consecutive odd integers such that 4 times the larger is 29 more than 3 times the smaller. Page | 26 Unit 3: Sequences Linear and Exponential Sequences A sequence can be thought of as an ordered list of elements. To define the pattern of the sequence, an explicit formula is often given, and for the purpose of this module, the first term is always found by substituting 1 into the variable for the explicit formula. A recursive sequence is defined by two pieces: o The first term of the sequence. A1 = 5 means Start at 5 o The rule for evaluating the sequence. An+1 = An – 7 translates to NEXT = PREVIOUS – 7 5, -2, -9, -16, - 23, -30, … The previous term and the next term of a sequence can be expressed in numerous ways – o a n 1 and a n , o a n and a n 1 , There are two types of sequences: 1. A sequence is an arithmetic sequence if there is a real number (known as the common difference) such that each term in the sequence is the sum of the previous term and . *Arithmetic sequences are also known as linear sequences. We know this sequence to be linear because it changes at a constant rate similar to a slope. The recursive formula for an arithmetic sequence is an1 an d . The explicit formula for an arithmetic sequence is an = a1 + d(n+1) 2. A sequence is a geometric sequence if there is a real number (known as the common ratio) such that each term in the sequence is a product of the previous term and . *Geometric sequences are also known as exponential sequences. We know this sequence to be exponential because it models exponential growth. The recursive formula for a geometric sequence is a n1 r a n . The explicit formula for a geometric sequence is an1 a1 r (n1) . Page | 27 55. Consider the following sequence: 9, 1, -7, -15, … i. Write an explicit formula that defines the sequence above. ii. Find the 38th term of the sequence. 56. Consider the recursive definition of sequence given by the formula an+1 = 5an and a1= 2 i. Explain what the formula means. ii. List the first 5 terms of the sequence. 57. Write the first 3 terms of the following sequences. Identify them as arithmetic or geometric. i. an+1 = an – 5 and a1 = 9 ii. an+1 = .5an and a1 = 4 iii. an+1 = 2 an and a1 = 4 iv. an+1 = an – 8 and a1 = 2 58. Identify each sequence as arithmetic or geometric. Explain your answer, and write an explicit formula for the sequence. i. 14, 11, 8, 5, … b. 2, 10, 50, 250, … Page | 28 Exponential Growth and Decay Exponential functions are modeled by the equation f(x) = ab x where “a” represents the initial value and “b” represents our growth/decay factor. o It is important to interpret what the “b” value of an exponential function is. If b > 1 the function represents a growth model. If 0 < b < 1 the function represents a decat model. 59. Given the following exponential functions determine: i. The initial value ii. If the following functions models growth or decay iii. The percent of change a. f(x) = 7347(1 + .17)x b. f(x) = 10(1 - .07) x c. f(x) = 57(1.32)x d. f(x) = 136(.86)x Page | 29
