Name: Teacher s Name: Estimated Test Date:

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1 Name: Teacher s Name: Estimated Test Date: Concept Page(s) Study Guide 2 3 Vocabulary 4 Constant and Average Rate of Change 5 6 Calculating Slope from Data and from a Graph 7 15 Graphing Slope-Intercept Form and Standard Form Writing Linear Equations Arithmetic Sequences Comparing Linear Functions (Graphs, Rules, and Tables) Scatterplots and Linear Regression Residual Plots Association and Correlation 41 Review P a g e

2 8.F.2 8.F.3 8.F.4 8.SP.1 8.SP.2 8.SP.3 Math 1 Unit 2: Linear Functions Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions.) Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or table of values. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. A-CED.2 Create and graph equations in two variables to represent linear, exponential, and quadratic relationships between quantities. A-REI.10 Understand that the graph of a two variable equation represents the set of all solutions to the equation. A-REI.11 Build an understanding of why the x-coordinates of the points where the graphs of two linear, exponential, and/or quadratic equations y = (x) and y = (x) intersect are the solutions of the equation (x) = (x) and approximate solutions using graphing technology or successive approximations with a table of values. A-SSE.1 F-BF.1 F-BF.2 F-IF.2 F-IF.3 F-IF.4 F-IF.5 F-IF.6 Interpret expressions that represent a quantity in terms of its context. a. Identify and interpret parts of a linear, exponential, or quadratic expression, including terms, factors, coefficients, and exponents. b. Interpret a linear, exponential, or quadratic expression made of multiple parts as a combination of entities to give meaning to an expression. Write a function that describes a relationship between two quantities. a. Build linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two ordered pairs (include reading these from a table). b. Build a function that models a relationship between two quantities by combining linear, exponential, or quadratic functions with addition and subtraction or two linear functions with multiplication. Translate between explicit and recursive forms of arithmetic and geometric sequences and use both to model situations. Use function notation to evaluate linear, quadratic, and exponential functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Recognize that recursively and explicitly defined sequences are functions whose domain is a subset of the integers, the terms of an arithmetic sequence are a subset of the range of a linear function, and the terms of a geometric sequence are a subset of the range of an exponential function. Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and maximums and minimums. Interpret a function in terms of the context by relating its domain and range to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Calculate and interpret the average rate of change over a specified interval for a function presented numerically, graphically, and/or symbolically. F-IF.7 F-IF.9 Analyze linear, exponential, and quadratic functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; rate of change; intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; and end behavior. Compare key features of two functions (linear, quadratic, or exponential) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions). 2 P a g e

3 F-LE.1 F-LE.2 F-LE.5 S-ID.6 S-ID.7 S-ID.8 S-ID.9 Math 1 Unit 2: Linear Functions Identify situations that can be modeled with linear and exponential functions, and justify the most appropriate model for a situation based on the rate of change over equal intervals. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters a and b in a linear function f(x) = ax + b or an exponential function g(x) = ab x in terms of a context. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a) Fit a least squares regression line to linear data using technology. Use the fitted function to solve problems. b) Assess the fit of a linear function by analyzing residuals. Interpret in context the rate of change and the intercept of a linear model. Use the linear model to interpolate and extrapolate predicted values. Assess the validity of a predicted value. Analyze patterns and describe relationships between two variables in context. Using technology, determine the correlation coefficient of bivariate data and interpret it as a measure of the strength and direction of a linear relationship. Use a scatter plot, correlation coefficient, and a residual plot to determine the appropriateness of using a linear function to model a relationship between two variables. Distinguish between association and causation. Students will know and be able to... determine the average rate of change for a given set of data. determine the slope of a linear function given two points or a graph. write an equation in slope-intercept form given a graph. interpret the meaning of the independent and dependent variables in the context of a linear function. can generate an arithmetic sequence for any number of terms given a rule or pattern can write the explicit rule and the recursive rule for a given arithmetic sequence. explain the meaning of the rate of change (slope) and y-intercept of a linear function in the context of a problem. use an appropriate linear model for a situation based on the rate of change over equal intervals. create and graph equations in two variables to represent a linear relationship between two quantities. choose an appropriate domain and range for a linear function. identify calculate, and interpret the x-intercepts and the y- intercepts of linear functions. identify if a linear function is increasing/positive or decreasing/negative compare two linear functions symbolically, graphically, verbally, and using tables\find the least squares regression line using technology compute and analyze residuals (linear and nonlinear models) use a model to interpolate and extrapolate predicted values determine the correlation coefficient using technology and determine the strength of the fit of the line distinguish between association and causation construct a scatter plot and draw a line of best fit Students will understand that the graph of a linear function represents all solutions. a linear function can be written in recursive form as an arithmetic sequence where the common difference C in the form NEXT = NOW + C corresponds to the slope m or a linear function expressed in slope-intercept form y = mx + b. a linear function can be constructed and interpreted given a graph, verbal description, table of values and/or a set of ordered pairs. not all data can be represented using a linear model. average rate of change of a linear function is always the same regardless of the interval. Essential Questions and Skills: How does rate of change relate to the slope of a linear function? How is slope calculated algebraically and graphically? Why is it important to be able to represent linear functions in a variety of ways? Why is important to compare rates of change and y-intercepts in various linear functions? How does an arithmetic sequence relate to a linear function? How can we model relationships between two quantities and use the model to make predictions? How does association differ from causation? How do we translate verbal ideas to the language of mathematics? 3 P a g e

4 Concept/Vocabulary Word Definition clustering constant of variation or constant of proportionality correlation coefficient when data seems to be gathered around a particular value or values It is the nonzero constant k in a direct variation equation. It is the constant ratio between the dependent and independent variables. a number that measures the strength and direction of the correlation between two variables direct variation a linear function defined by an equation of the form y = kx, where k 0 domain explicit rule line of best fit linear equation linear model the set of all inputs or x-values a rule that can be used to find any term in a sequence without having to calculate any other term It is a straight line through the data on a scatterplot that comes closest to all the data. One such line is a regression line that uses least-squares linear regression. an equation whose graph forms a straight line using a linear representation for a set of data linear relationship negative association negative slope no association has a graph that forms a straight line and therefore depicts a constant or average rate of change one variable s values increase as the other variable s values decrease the slope of a line that has decreasing y-values as the x-values increase the variables values seemingly have no relationship to each other nonlinear association not in a straight line point slope form the equation of a straight line in the form y y1 = m(x x1) where m is the slope of the line and (x1, y1) are the coordinates of a given point on the line positive slope positive association the slope of a line that has increasing y-values as the x-values increase one variable s values increase as the other variable s values increase average rate of change recursive rule residual scatterplot slope slope-intercept form the difference between two output values divided by the difference between the corresponding input values. a rule that can be used to create a sequence in which the preceding term in the sequence is used to find the next term The difference between the observed y-value and the predicted y-value a graph with points plotted to show a possible relationship between two sets of data a measure of the steepness of a line on a graph; the rise divided by the run; the constant rate of change of a linear function ; the change in y-values as the x-values increase by 1 the form of a linear equation described by the equation y = mx+b where m is the slope of the line and b is the y-intercept. standard form of a linear equation the form of a linear equation described by the equation Ax + By = C, where A, B, and C are typically integers, A and B are not both zeros, A is nonnegative, and A, B, and C have no common factors other than 1 x-intercept the x-coordinate of the point where the line crosses the x-axis ; the value of x when y = 0 y-intercept the y-coordinate of the point where the line crosses the y-axis ; the value of y when x = 0 4 P a g e

5 Rate of Change and Average Rate of Change Math 1 Unit 2: Linear Functions In this lesson you will estimate the rate of change from a graph. calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Every day we deal with quantities expressed as ratios: miles per gallon of gas, cost per kilowatt of power, miles per hour that a car is travelling. When working with functions that relate two quantities such as miles and gallons or cost and kilowatts or miles and hours, we refer to these ratios as rate of change. Rate of change tells us how much one quantity is changing with respect to another quantity. For example, a speed of 60 mph tells us that a vehicle travels 60 miles for each hour it is driven. Some rates of change are constant, and others are not. For example, if a car travels from one city to another, it does not normally travel at a constant rate. The car will speed up or slow down depending on traffic, or may stop for a period of time so the driver and passengers can grab a bite to eat. When the rate is not constant, we often look at the average rate of change. The average rate of change tells us how much one quantity changes with respect to another quantity over a specified interval. So if the car travels 150 miles in 3 hours, we can say that the average rate of change (or speed) that the car travelled was 50 miles per hour. Typically, when looking at functions, we are interested in the rate of change of a function s output. Below the following graphs are three stories about walking from your locker to your class. Two of the stories below correspond to two of the graphs. Match the stories with a graph. Draw your own graph to matches the third story. 1. I started to walk to class, but I realized I had forgotten my notebook, so I went back to my locker and then I went quickly at a constant rate to class. 2. I rushed from my locker to get to class when I realized I wasn t really late, so I slowed down a bit. 3. I started walking from my locker at a steady, slow, constant rate to my class, and then, realizing I was late, I ran the rest of the way at a steady, faster rate. 5 P a g e

6 Interpretations of Graphs - Practice Math 1 Unit 2: Linear Functions 1. Ian and his three friends were sitting on a deck and eating sunflower seeds. Each person had a bowl with the same amount of seeds. The graphs below all show the amount of sunflower seeds remaining in the person s bowl over a period of time. Write sentences that describe what may have happened for each person. Focus on describing rates that are constant and not constant. a) b) c) d) Indicate which graph matches the statement. Give reasons for your answer. 2. A bicycle valve s distance from the ground as a boy rides at a constant speed. a) b) c) d) 3. A child swings on a swing, as a parent watches from the front of the swing. 6 P a g e

7 Calculate the average rate of change between each pair of two points. Cross out each box in the rectangle below that contains the correct answer. Print the remaining letters to find out what you call a duck that steals. 7 P a g e

8 Slopes of Lines Math 1 Unit 2: Linear Functions The constant rate of change of a line is called slope. The slope of a direct variation is called the. Normally, the variable for a direct variation s slope is. If the line is not a direct variation, the variable used for slope is usually. All lines have one of 4 slopes: positive, negative, zero, and undefined. Only lines with undefined slopes aren t functions. 1. State whether the following slopes go up or down (left to right). a. -3 b. 1 2 c. 5 2 d. 9 e. -10 Calculating slope from two given points. The equation to calculate the slope, m, between two given points {(x 1, y 1 ), (x 2, y 2 )} is m = y 2 y 1 x 2 x 1 Use the equation to calculate the slope of the line that passes through each pair of points. 2. (-2, -2), (4, 1) 3.(-5,3), (2,1) 4. (3, 5), (-1, 4) 5. (3, 5), (1, 4) 6. (5, -3), (2, 1) Calculating slope given points on a line. 7. What is the slope of this line? 8. Explain in your own words how you find the slope of a line given the graph. 9. Explain in your own words what the slope means. 8 P a g e

9 10. State the slope of the line. 11. State the slope of the line. 12. All horizontal lines have a slope of. All vertical lines have an slope. Graphing lines when given a point on the line and the slope. Example: The line passes through (2,0) and has a slope of 3 1. Graph the ordered pair (2, 0). 2. From the known point, use the slope to calculate a second point. 3. Plot a point at this location. 4. Use the slope to find a third point. This step is optional. 5. Draw a straight line through the points. 13. point = ( -2,4) ; slope = -2/3 14. Point (0,-4) and slope = 0 9 P a g e

10 Calculate missing coordinates of points on a line by using the slope. Example: The slope of a line that goes through the points (r, 8) and (4, 2) is 4. Find r. To solve this, substitute the given information into the formula. m = y 2 y 1 x 2 x 1 Calculate r. 15. m= 3 2 and the line goes through (r,6) and (10,-3). 16. m = 3 4 and points (3,4) and (-1,r) Graph each ordered pair. Calculate the slope of the line. 17. (1,2) and (3, 6) 18. (2,6) and (4, 0) 19. (3, 2) and (1, 4) 20. (2,5) and (1, 2) 10 P a g e

11 Calculate the slope of the line passing through the given points. 21. (7, 4) and (3, 2) 22. (8, 8) and (-4, -4) 23. (3, 2) and (5, 6) 24. (0, -1) and (1, -6) Graph the line given the slope and a point on the line 25. Slope of 1 and point (7, 8) 26. Slope of -3 and point (2, 4) The slope of a line that goes through the points (x, -5) and (-1, -4) is 1. Find x The slope of a line that goes through the point (1, 4) and (3, y) is 3. Find y. 11 P a g e

12 Slopes of Lines Math 1 Unit 2: Linear Functions The constant rate of change of a line is called slope. The slope of a direct variation is called the. Normally, the variable for a direct variation s slope is. If the line is not a direct variation, the variable used for slope is usually. All lines have one of 4 slopes: positive, negative, zero, and undefined. Only lines with undefined slopes aren t functions. 1. State whether the following slopes go up or down (left to right). a. -3 b. 1 2 c. 5 2 d. 9 e. -10 Calculating slope from two given points. The equation to calculate the slope, m, between two given points {(x 1, y 1 ), (x 2, y 2 )} is m = y 2 y 1 x 2 x 1 Use the equation to calculate the slope of the line that passes through each pair of points. 2. (-2, -2), (4, 1) 3.(-5,3), (2,1) 4. (3, 5), (-1, 4) 5. (3, 5), (1, 4) 6. (5, -3), (2, 1) Calculating slope given points on a line. 7. What is the slope of this line? 8. Explain in your own words how you find the slope of a line given the graph. 9. Explain in your own words what the slope means. 12 P a g e

13 10. State the slope of the line. 11. State the slope of the line. 12. All horizontal lines have a slope of. All vertical lines have an slope. Graphing lines when given a point on the line and the slope. Example: The line passes through (2,0) and has a slope of 3 1. Graph the ordered pair (2, 0). 2. From the known point, use the slope to calculate a second point. 3. Plot a point at this location. 4. Use the slope to find a third point. This step is optional. 5. Draw a straight line through the points. 13. point = ( -2,4) ; slope = -2/3 14. Point (0,-4) and slope = 0 13 P a g e

14 Calculate missing coordinates of points on a line by using the slope. How many different methods can you think of that will allow you to calculate the missing coordinate? Calculate r. 15. m= 3 2 and the line goes through (r,6) and (10,-3). 16. m = 3 4 and points (3,4) and (-1,r) Graph each ordered pair. Calculate the slope of the line. 17. (1,2) and (3, 6) 18. (2,6) and (4, 0) 19. (3, 2) and (1, 4) 20. (2,5) and (1, 2) 14 P a g e

15 Calculate the slope of the line passing through the given points. 21. (7, 4) and (3, 2) 22. (8, 8) and (-4, -4) 23. (3, 2) and (5, 6) 24. (0, -1) and (1, -6) Graph the line given the slope and a point on the line 25. Slope of 1 and point (7, 8) 26. Slope of -3 and point (2, 4) The slope of a line that goes through the points (x, -5) and (-1, -4) is 1. Find x The slope of a line that goes through the point (1, 4) and (3, y) is 3. Find y. 15 P a g e

16 Slope-Intercept Form of Linear Functions y=mx + b is called slope-intercept form. m is the constant rate of change (slope) and b is the value of y when x = 0 (y-intercept). The constant added to y=mx translates the function up if the constant is greater than zero and down if the constant is less than zero. If a diagonal line doesn t pass through the origin, it can be written as a vertical translation of a direct variation. The slope determines how steep the line is. As m approaches zero, the line becomes less steep. As the value of m gets further from zero, the line becomes more steep. Lines with positive slopes are increasing functions. Lines with negative slopes are decreasing functions. To graph any function, substitute in a domain value into the function rule to calculate the corresponding range value. Graph the point, and continue this process until the function rule is graphed. Because most functions can have any real number as a domain value, it is important to learn the characteristics of functions in order to graph them more quickly. Using Patterns to Graph Linear Functions Written in Slope- Intercept Form Example: Graph y = 2x 4 1. Identify the slope (value and sign) and y-intercept: Slope y-intercept: 2. Graph the y-intercept on the coordinate plane. 3. Next, count the slope (change in y as x increases by 1) from the y-intercept and place a second point. 4. Use the slope again to plot at least one other point. This step is optional. 5. Check to make sure that both points are solutions of the original equation. If they are correct, draw the line. Representing Linear Function Rules as Points and Graphs For each linear function, calculate the y-intercept and one other point. Graph the line. 1. Graph y = 1 x Graph y = 3x P a g e

17 Direct Variation Review Math 1 Unit 2: Linear Functions Direct Variation a function of the form, where k 0. k is the. It is the coefficient of x. The variables y and x are said to When graphed, the y-intercept of a direction variation is. Any two points (except for ) from a direct variation can be used to create a. Example 1: Example 2: y = 3x y = x Constant of Variation: Constant of Proportionality: 2 points it passes through: 2 points it passes through: State whether or not the equation is direct variation. If it is, solve the equation for y and calculate the constant of variation. 1) y = 2x 2) 2x + 5y = 1 3) -12x = 6y 4) y + 8 = -x 5) 5x 6y = 0 6) -x = 10y Write an equation of direct variation that includes the given point: 7) (1, 5) 8) (-8, 10) 9. (-6, 1) 10. (3, -4) Assume that y varies directly with x: 11) If y = 28 when x = 7, find x when y = 52 12) If y = 2.5 when x = 0.5, find x when y = 20 13) If y = 4 when x = 12, find y when x = ) If y = when x = ¼, find y when x = ) What are some real-world examples of direct variations? 17 P a g e

18 Direct Variation Applications 1. In 1852, Henri Giffard built the first airship successfully used for transportation. It had a volume of 88,000 cubic feet and could support 34,000 pounds. The Graf zeppelin II, built in 1937 had a volume of 7,063,000 cubic feet, making it one of the two largest airships ever built. The weight an airship can support varies directly with its volume. How much weight could the Graf zeppelin II support? 2. The income at a store varies directly as the advertising budget. When the owners spent $2300 per month to advertise, their monthly gross income was $120,000. If the owners increase their advertising budget to $5000 per month, how much gross monthly income can they expect? 3. The length that a spring will stretch S varies directly with the weight w attached to the spring. If a spring stretches 11 inches when a 20 pound weight is attached, how far will it stretch when a 10 pound weight is attached? 4. State the constant of variation (constant of proportionality) for each equation. 1 4 a. y x b. y = 2.5x 5. The perimeter of a square with side length s is modeled by. Do the side length and the perimeter have direct variation? 6. The variables x and y vary directly. Use the given values to write an equation that relates x and y. a. x = 5, y = b. x =, y = 6.8 x = 10, y = 6 x = 4, y = Sound travels about 12.4 miles in one minute. How long, in seconds, does it take sound to travel 30 miles? 8. If weight varies directly with gravity and a person who weighs 120 pounds on Earth weighs 57 pounds on Mars, How much does a person who weighs 210 pounds on Earth weigh on Mars? 18 P a g e

19 The standard form of a linear function gives you another option for graphing equations. Instead of using properties of equality to solve for y and write the equation to slope-intercept form, you can use standard form to quickly solve for the x- and y-intercepts, then graph. Many real-world problems with a constant rate of change can be modeled by standard form. Standard form is Ax + By = C where A, B, and C are typically integers, A is non-negative, and A, B, and C only share a common factor of Calculate the x-intercept and y-intercept of the line 2x + 3y = 6. Use the intercepts to graph the equation. Find the x-intercept: Find the y-intercept: Use the properties of equality to rewrite the equation in slopeintercept form. 2. Calculate the x-intercept and y-intercept of the line x y = 3. Use the intercepts to graph the equation. Find the x-intercept: Find the y-intercept: Use the properties of equality to rewrite the equation in slope-intercept form. 19 P a g e

20 Representing Linear Graphs as Function Rules and Points 1. For the line graphed below, complete the table of values and write the equation. Table of values: x y Equation in Slope-Intercept Form: Equation in Standard Form (A is a whole number. B & C are integers.) Representing Points on a Line as Graphs and Function Rules 2. From the table, draw the graph and write the equation of the line in slope-intercept form. Graph: x y Equation in Slope-Intercept form: Equation in Standard Form (A is a whole number. B & C are integers.) 20 P a g e

21 For each table, state whether or not the relation is a function and whether or not it is linear. If it is linear, write the equation of the line that passes through the points in slope-intercept form. 1. x y Write the equation of the line given two points in slope-intercept form. 5) (1, 8), (4, 17) 2. x y ) (2, -4),(6, -6) 3. x y ) (1, 3), (4, 15) 4. x y ) (3, 4), (6, 6) 9) (29, 21.25), ( 7, 5.75) 21 P a g e

22 For each set of ordered pairs, write a linear function rule in each of the three forms. Check each equation. 1. { (6,3), ( 12, 9)} 2. {( 5, 22), ( 9, 34)} 3. {(0, 2), ( 5, 0)} 4. {( 3, 5), (0,2)} 5. {( 5, 5), (3, 3)} 6. Describe how to quickly graph a line written in A) slope-intercept form B) point-slope form C) standard from 7. How do you write the equation of a horizontal line? In which form is it written? Is it a function? 8. How do you write the equation of a vertical line? In which form is it written? Is it a function? 22 P a g e

23 1. Identify each table of values as linear or nonlinear and explain. x f(x) x y x f(x) Write an equation in slope-intercept form for the following relations. 2. x y x y x y (1, 8) (4, 17) 6. (2, -4) (6, -6) 7. (1, 3) (4, 15) 8. (3, 4) (6, 6) 23 P a g e

24 For each of the following, identify the form of the given linear function rule (Standard, Slope-Intercept, or neither). Write it in each of the 2 forms. Check your 2 equations by calculating 2 points on the line and checking to make sure that those 2 points are solutions to both of the equations. When writing in standard form, use only integers which share no factors other than 1, and the coefficient of x should be non-negative. 1. y = 2 3 x 2 2. y 3 = 5 (x + 2) x 3y = 6 4. y + 2 = 4(x + 1) 5. 5x + 2y = y = 4x 2 7. Which form are vertical lines (x = a) and horizontal lines (y = b) written in? Is there more than one correct answer? 24 P a g e

25 Arithmetic Sequences Notes & Practice Math 1 Unit 2: Linear Functions An arithmetic sequence goes from one term to the next by always adding (or subtracting) the same value. The number added at each stage of an arithmetic sequence is called the "common difference" d, because if you subtract successive terms, you'll always get this common value. Use what you learned in Unit 1 along with unit 2 to look for patterns between arithmetic sequences and linear functions. 1) 5, 8, 11, 14,.. Start, Next = Now 2) 7, 3, -1, -5,. Start, Next = Now Table Recursive Rule Explicit Rule Graph (y=mx +b) 3) 4) x f(x) ) Are these patterns arithmetic? How do you know? 6) Explain how the common difference in a recursive rule relates to the explicit rule for arithmetic sequences. 7) Explain how the start value in a recursive rule relates to the explicit rule for arithmetic sequences. 8) Are the graphs linear or nonlinear? Why? 25 P a g e

26 Studying Arithmetic Sequences 1) Term number Total # of shapes Let t = term number ; Let s = total # of shapes Rule: How many shapes will be in the 102 nd term? What term will have 314 shapes in it? 2) Term number Total # of shapes Let x = term number ; Let y = total # of shapes Rule: How many shapes will be in the 34 th term? What term will have 77 shapes in it? 3) Design number Total # of shapes Let d = design number; Let y = total # of shapes Rule: How many shapes will be in the 10 th term? What term will have 135 shapes in it? 26 P a g e

27 4) Patio number Total # of white tiles Let p = patio number; Let w = total # of white tiles Rule: How many white tiles will be in patio 50? What patio will have 52 white tiles on it? 5) # of black tiles # of white tiles Let b = # of black tiles ; Let w = # of white tiles Rule: How many white tiles will there be if there are 7 black tiles? How many black tiles will there be if there are 19 white tiles? 27 P a g e

28 Money Raised ($) Math 1 Unit 2: Linear Functions Comparing Functions 1. Several students who are participating in a 10k walk-a-thon to raise money for charity need to decide on a plan for sponsors to pledge money. Jeff thinks $1.50 per kilometer would be an appropriate pledge. Rachel suggests $2.50 per kilometer because it would bring in more money. Annie says that if they ask for too much money, people won t agree to be sponsors; she suggests they ask for a donation of $4 and then $0.75 per kilometer. Jeff s Plan Rachel s Plan Annie s Plan Distance Money Raised ($) Distance Money Raised ($) Distance Money Raised ($) 0 km 1 km 2 km 3 km 4 km 5 km 6 km 7 km 8 km 9 km 10 km 0 km 1 km 2 km 3 km 4 km 5 km 6 km 7 km 8 km 9 km 10 km 0 km 1 km 2 km 3 km 4 km 5 km 6 km 7 km 8 km 9 km 10 km A. In the tables provided, calculate the range values for the given domain values. B. On the graph paper provided, graph all three pledge plans on the same coordinate grid. Use different colored lines to connect the points. Be sure to include a title, label the axes with intervals marked, and a key to indicate each plan. 28 P a g e Distance (km)

29 C. For each of the three pledge plans, use words to describe the relationship between the money earned and the distance walked. Jeff s Plan- Rachel s Plan- Annie s Plan- D. For each of the different pledge plans: a) Write a NOW-NEXT rule. b) Write an explicit function. Let p = the money ($) owed under each pledge plan. Let d = distance the student walks (km). Jeff s Plan: a) b) Rachel s Plan: a) b) Annie s Plan: a) b) E. Study the rate of change (slope) and the y-intercept in each of the four representations (table, graph, description, and rule). a) Give the rate of change (slope) in each of the plans. Was one representation easier than the others to find rate of change? Which one and Why? b) What was the y-intercept in each plan and was one representation easier for finding it? 29 P a g e

30 F. Analyze the problem by answering the following questions: 1) Describe how increasing the amount of pledge per kilometer affects the tables, the graphs and the equations? 2) Describe what is different about Annie s plan. 2. The graph below shows the relationship between weekly profit and the number of customers per week for Skate World Roller Rink. Skate World Weekly Profit 4,000 3,000 Weekly Profit (in dollars) 2,000 1, ,000 1,200-1,000-2,000 Number of Customers per Week A. Determine the slope and y-intercept of the line that fits this data pattern. B. Explain what the slope and y-intercept of the line tell you about the relationship between Skate World profit and number of customers per week. C. If Skate World reached maximum capacity during each skating session for a week, admissions for that week would total 2,400 customers. Estimate the rink s profit in this situation. Explain your reasoning. D. What is the meaning of the ordered pair (400, 0) in terms of the number of customers and weekly profit? 30 P a g e

31 3. The table below show data from an investigation entitled Taking Chances. Number of Trials Cumulative Profit (in $) A. Is a linear model is reasonable for these data? Explain. B. Is cumulative profit an exact linear function of the number of trials? Explain why or why not. C. Write a NOW-NEXT rule to represent the relationship between number of trials and cumulative profit. D. Write a function rule in the form y = mx + b. E. What is the coefficient of the independent variable in your model in Part D? What does it tell you about the relationship between the cumulative profit and number of trials? F. How many trials will need to be taken in order to have a cumulative profit of $74? Explain your reasoning. 3 Representations of Functions: Equations, Graphs, and Tables. For each of the following, create the missing representations of the given linear function x + 2y = 6 7. Explain how you can check your answers to problems like P a g e

32 8. On Penny s 15 th birthday, her grandmother gave her a large jar of quarters. Penny decided to continue to save quarters in the jar by using a pattern. Every few months she counts her quarters and records the number a table like this one. Predict how many quarters she ll have on her 18 th birthday. Penny s Savings Number of months since Penny s 15 th birthday Number of quarters a) Figure out Penny s pattern. Fill in the table above with the missing number of quarters. b) Identify the independent variable. Identify the dependent variable. c) Does there appear to be a linear function rule that could explain Penny s pattern? Explain your reasoning. d) Use a calculator to graph the 4 known data points. After you analyze them, add the 4 points that you calculated to the calculator s graph. Do the points support your answer to part c? e) Find the slope of the data and explain the real-world meaning of the slope as related to the data described in the table. f) Write a NOW-NEXT rule to show how the number of quarters Penny saves changes with each month she saves. g) Use the slope you found in part e to write an equation of the form y = mx + b. h) What is the real-world meaning of the y-intercept? i) Use your equation to predict how many quarters Penny will have on her 18 th birthday. Adapted from Discovering Algebra-2 nd Edition Chapter 4 32 P a g e

33 Scatterplots, Correlation and Association Positive correlation (positive association) when one variable increases the other increases Negative correlation (negative association) when on variable increases the other decreases No correlation (no association or zero association) no relationship between the variables Identify the type of correlation in each scatterplot below. What type of correlation would you expect the following relationships to have? 1. The weight of a cat and how much it eats. 2. The size of your feet and how many siblings you have. 3. The amount of time spent watching TV and a student s grade in math. 4. The scatter plot shows the relationship between games played and tickets won. Which graph below represents the line of best fit? 5. The scatter plot shows the relationship between hours spent shopping and money. Which graph below represents the line of best fit? 33 P a g e

34 6. Look at the graph below comparing height and weight. Draw a line of best fit and answer the following. a. How much would you expect someone who is 172 centimeters to weigh? b. How tall would you expect someone who weighed 85 kg to be? Linear Regression of Scatter Plots to Calculate the Line of Best Fit Many calculators use complex algorithms that find a precise line of fit called the best-fit line. One algorithm is called linear regression. It is sometimes called least squares regression. To enter the data: Calculator Steps: LIST 2nd L 1 is independent variable; L 2 is dependent variable LIST CALC ; 5: LinReg ; The calculator will display LinReg(ax+b) 2 nd y1. APPS 2: Y-VARS ; 1: Y 1 Enter This will calculate the line of best fit and store it under function Your calculator can also compute a number called the correlation coefficient. This number will tell you if your correlation is positive or negative and how closely the linear equation is modeling the data. The closer the correlation coefficient is to 1 or 1, the more closely the equation models the data. To turn the correlation coefficient on 2nd : PRGM DiagnosticOn Enter 1. If the correlation coefficient is close to 1 or -1, is the correlation strong or weak? 2. If the correlation is closer to zero (between -0.5 and 0.5), is the correlation strong or weak? 3. If the scatterplot appears random, there is correlation. 4. If the correlation coefficient is positive, the slope of the line of best fit will be. 5. If the correlation coefficient is negative, the slope of the line of best fit will be. We will often need to interpret the slope and y-intercept in the context of the problem. When contextualizing a function rule, focus on the topic of the data, whether the correlation is positive or negative, the value of the slope, and the units of the independent and dependent variables. EXAMPLE: The average lifespan of American women (y) has been tracked, and the model for the data is y = 0.2t + 73, where t = 0 corresponds to Interpret the equation (contextualize): 34 P a g e

35 Real World Example 1: Box Office The table shows the amount of money made by movies in the United States. Let x = the number of years after Use a graphing calculator to write an equation for the best-fit line for that data. Round the slope and y-intercept to the nearest hundredth. 1. What is the equation of the best fit line using the graphing calculator. 2. r = Describe the fit. 3. Interpret the slope. 4. Interpret the y-intercept. 5. EXTRAPOLATION: Use the equation and the table in the graphing calculator to predict what the box office income will be in State your answer as a complete sentence. 6. INTERPOLATION: Use the equation and the table in the graphing calculator to calculate what the expected box office income was in How does the compare to the actual box office income given in the table? What is the difference? 35 P a g e

36 Real World Example 2: Cleaning Services Math 1 Unit 2: Linear Functions Data was collected on the costs of various cleaning companies. Round your answers to the nearest tenth. 1. Calculate the slope of the line of best fit. 2. Calculate the y-intercept of the line of best fit. 3. Calculate the correlation coefficient. 4. Explain what the slope means (contextualize). Length of Visit (in Cost of Visit (in hours) dollars) Explain what the y-intercept means (contextualize). 6. Explain what the correlation coefficient means. How good is the line of best fit? 7. Interpolate: According to the line of best fit, about how much could you expect to pay if a cleaning company cleaned your house for 5 hours? 8. Extrapolate: According to the line of best fit, about how much could you expect to pay if a cleaning company cleaned your house for 10 hours? 9. Interpolate: According to the line of best fit, about how much could you expect to pay if a cleaning company cleaned your house for 4 hours? 10. Calculate the residual. How much does your answer to #9 differ from the actual data collected? 11. Based on this worksheet, what do you think interpolate and extrapolate mean? 36 P a g e

37 Linear Regression and Lines of Best Fit Math 1 Unit 2: Linear Functions 1. Use the data on neighborhood children to write an equation (using your calculator) and then use your equation to answer the questions. a. Equation b. How tall would you expect a two year old to be? c. How old would you expect a child who is 47 inches tall to be? Above is a scatterplot showing the relationship between time since noon and money left for a person gambling. 2. Use two points that fall on the line to write an equation for your line of best fit. 3. What does the y-intercept mean in this problem? 4. What does the slope mean in this problem? 5. Use your equation to predict when the gambler runs out of money. 6. At 10:00pm the gambler says, Hey, I m on a roll! I ve been making money for the last two hours. Is this necessarily true? How would you advise the gambler? 7. Confirm your answer by using your line of best fit to estimate how much the gambler will have left after 14 hours of gambling. 8. What if the gambler kept going for 24 hours? What would be the dollar value? What does this mean in the context of this problem? 9. Is a linear function an appropriate way of dealing with this situation? Why or why not? 37 P a g e

38 Residuals: observed value predicted value Math 1 Unit 2: Linear Functions Press "List" Highlight L3 Press "2nd" and "List" Choose RESID (#7). It should now read L3 = RESID. Press Enter. The third list should now be titled "RESID" and there should be data in the list. Press "2nd" and "y=", and turn the first plot on. Move the cursor to Ylist so that the cursor is blinking on "L2". Change it to "L3" by choosing "L3" under "2nd" "List" Plot the graph. If necessary, Use "Zoom" and "ZoomStat" to make your graph fit the screen. 1. What are residuals? 2. What are some examples of correlation without causation? 3. What are some examples of correlation with causation? Would they be positive or negative correlation? Residual Plots A residual plot is a graph that shows the difference between the actual data (what is provided through a table or graph) and the predicted data (what the model says should happen). The independent variable is graphed on the horizontal axis and the residual value (actual predicted) is graphed on the vertical axis. When using linear regression, if the residual plots are randomly scattered around the horizontal axis, a linear model is the best choice to model the data. If the residual plot shows a pattern, and does not appear to be random or scattered, a non-linear model would most likely be a better fit. Look at the residuals from the linear regression of 3 different sets of data. The first one has residuals scattered around the horizontal axis, so the data is best modeled by a linear function. The last two residuals show patterns, so non-linear functions should be used to calculate a curve of best fit. 38 P a g e

39 Calculating and Analyzing Residuals Example 1: The data below shows the number of active woodpecker clusters in the DeSoto National Forest. Year Active Clusters Enter the data into your calculator and calculate the line of best fit. Let x represent the number of years after What is the value of the correlation coefficient? What does this tell you about how good of a fit the linear equation is? 3. Using your equation from question 1, calculate the predicted number of active woodpecker clusters. Year Predicted Active Clusters 4. To find the residual plots, we need to find the difference between what actually happened (original table) and what is predicted to happen (table from #3). Year Residual Value 5. Construct a residual plot. On the horizontal axis will be the independent variable. On the vertical axis will the residual value. 6. Would you describe the residual plot as scattered around the horizontal axis and random or do you see a pattern? Do you think a linear model is best? 39 P a g e

40 Example 2: Math 1 Unit 2: Linear Functions The table below shows the population data from the U.S. Census. Round all answers to the nearest hundredth. 7. Use linear regression to calculate the equation of the line of best fit. Let x represent the number of years since Use the model, with rounded values, to calculate anticipated values for the same years. 9. Calculate the residuals and create a residual plot. 10. Based on the residual plot, do you think a linear model is best? Justify your response. Year Population (in millions) Year 1900 Population (in millions) Residuals Use the graphing calculator to calculate the equation of the curve of best fit using exponential regression. 12. Use the graphing calculator to calculate the equation of the curve of best fit using quadratic regression. 13. Of the three models, which model do you think would be best? Justify your answer using complete sentences. 14. Contextualize the slope and the y-intercept from the equation for the line of best fit in # P a g e

41 Correlation and Causation Math 1 Unit 2: Linear Functions A correlation is a measure or degree of relationship between two variables. A set of data can be positively correlated, negatively correlated or not correlated at all. As one set of values increases the other set tends to increase then it is called a positive correlation. As one set of values increases the other set tends to decrease then it is called a negative correlation. If the change in values of one set does not affect the values of the other, then the variables are said to have "no correlation" or zero correlation". A causal relation between two events exists if the occurrence of the first causes the other. The first event is called the cause and the second event is called the effect. A correlation between two variables does not imply causation. On the other hand, if there is a causal relationship between two variables, they must be correlated. Example 1: A study shows that there is a negative correlation between a student's anxiety before a test and the student's score on the test. In other words, the higher the student s anxiety, the the test score. Causation: Does anxiety cause a student to earn a low test score? Example 2: A study shows that there is a positive correlation between the number of hours a student spends studying and the student s score on a test. In other words, the more hours spent studying, the the test score. Causation: Does more studying result in a higher grade? 1. Decide if each of the following data would likely show correlation. If so, is it positive or negative correlation and is it causation or is it correlation without causation? a. a young child's weight and the number of words in her vocabulary b. the number of minutes spent exercising and the amount of calories burned c. the amount of cars a salesperson sells and how much commission she makes d. The number of cold, snowy days and the amount of hot chocolate sold at a ski resort e. The number of miles driven and the amount of gas used f. the weight of the puppy in New York and the retail price of snowshoes in Alaska Enter the data. Use linear regression to calculate the line of best fit. 2. What is the slope? What does it mean? 3. What is the y-intercept? What does it mean? 4. What is the correlation coefficient? What does it mean? Linear regression is one way to calculate the line of best fit. It calculates the slope and the y-intercept of the line that will have the smallest sum of the squares of the residuals. The given formulas are for educational purposes only. You will not need to calculate the equation of the least squares regression line by hand. 41 P a g e

42 Unit 4 Practice Assessment 1. Calculate the slope of the line that passes through (1, 3) and ( 2, 4). Clearly show your work. 2. State the slope and y-intercept of the line: y = 3 2 x 1 slope: y-intercept: 3. A line passes through (1, 1) and its slope is 4. Write the equation of the line in slope-intercept form. 4. Calculate the slope and the y-intercept of the line. Show your work. x + 2y = 8 slope: y-intercept: 5. Write the equation of the line in standard form and point slope form that passes through (1, 4) and has a slope of Write the equation of the direct variation that passes through (2, 5). 7. A direct variation passes through (3, 8). Another point on the line is (x, 4). Calculate the value of x. The line also passes through (9, y). Calculate the value of y. 8. Graph y = x 1 y X 9. Write the slope-intercept form of the line that passes through (1, 3) and (3, 1). 10. Write y = 5x + 2 in standard form. 11. Write the standard form of the line that passes through ( 1, 2) and (1, 3). 42 P a g e