4.1 Writing Equations in Slope-intercept Form Notes Vocabulary Definition Picture Linear model Example 1: Write an equation with the given slope Monitoring Progress and y-intercept. 1. Slope = 7 y-intercept = 2 a. slope = -3 y –intercept ½ b. slope = 0 y-intercept -2 2. slope = 1/3 y-intercept = -1 Example 2: Write an equation of each in slope- Monitoring Progress intercept form 3. 4. a. b. Example 3: Write an equation of each line that Monitoring Progress passes through the given points 5. (0, -2) (4, 10) 6. (4, -1) (0, 2) a. (-3, 5) (0, -1) b. (0, -5) (8, -5) Example 4: Write an function f with the values Monitoring Progress f(0) = 10 and f(6) = 34 7. g(0) = 9 and g(8) = 7 Example 5: Monitoring Progress 8. 4.2 Writing Equations in Point-Slope Form Notes Example 1: Write an equation in point-slope Monitoring Progress form of the line that passes through the point 1. (3, -1); m = -2 (-8, 3) and has a slope of ¼ 2. (4, 0); m = -2/3 Example 2: Write an equation in slope-intercept Monitoring Progress form of the line shown. 3. (1, 4) (3, 10) 4. (-4, -1) (8,-4) Example 3: Write a linear function f with the Monitoring Progress values f(4) = -2 and f(8) = 4 5. g(2) = 3 and g(6) = 5 Example 4: Monitoring Progress 6. 4.3 Writing Equations of Parallel and Perpendicular Lines Notes Vocabulary Definition Picture Parallel lines Perpendicular lines Example 1: Determine which of the lines are Monitoring Progress parallel. 1. Line a passes through (-5, 3) & (-6, -1) Line b passes through (3, -2) & (2, -7) are the lines parallel? Example 2: Write an equation of the line that Monitoring Progress passes through (5, -4) and is parallel to the line 2. a. Passes through (-4, 2) Parallel to y = 1/4x + 1 y = 2x + 3 b. Passes through (3, 1) Parallel to y = 3x + 1 Example 3: Determine which of the lines if Monitoring Progress any are parallel or perpendicular. 3. Line a: 2x + 6y = -3 Line b: y = 3x - 8 Line a: y = 4x + 2 Line b: x + 4y = 3 Line c: -8y -2x = 16 Line c: -6y + 18x = 9 Example 4: Write an equation of the line that Monitoring Progress passes through (-3, 1) and is perpendicular to the line y 4. a. passes through (-3, 5) perpendicular to y = -3x -1 = 1/2x + 3 b. passes through (4, -3) perpendicular to y = 4x - 1 Example 5: Monitoring Progress 5. In example 5 a boat is traveling parallel to the shore line and passes through (9, 3). Write an equation that represents the path of the boat. 4.4 Scatter Plots and Lines of Fit Notes Vocabulary Definition Picture Scatter plot Correlation Line of fit Positive Correlation Negative Correlation No Correlation Example 1: The scatter plot shows the amounts x Monitoring Progress (in grams) of sugar and the numbers y of calories in 10 smoothies. 1. How many calories are in the smoothie a. How many calories are in the smoothie that contains that contains 51 grams of sugar? 56 grams of sugar? b. How many grams of sugar are in the smoothie that contains 320 calories? c. What ends to happen to the number of calories as the 2. How many grams of sugar are in the number of grams of sugar increases? smoothie that contains 250 calories? Example 2: Tell whether the data show a Monitoring Progress positive, a negative or no correlation. 3. Make a scatter plot of the data. Tell where the data show a positive, a negative, or no correlation. Example 3: The table shows the weekly sales of a Monitoring Progress DVD and the number of weeks since its release. Write 5. The following data pairs show the monthly income x (in dollars) an equation that models the DVD sales as a function of and the monthly car payment y (in dollars) of six people: (2100, 410) the number of weeks since its release. Interpret the (1650, 315) (1950, 405) (1500, 295) (2250, 440) (1800, 375) slope and y-intercept of the line of fit. 4.5 Analyzing Lines of Fit Notes Vocabulary Definition Picture Residual Linear regression Line of best fit Correlation coefficient interpolation extrapolation causation Example 1: in Example 3 in section 4.4 the equation Example 2: The table shows the age x and salaries y y = -2x + 20 models the data in the table shown. Is the (in thousands of dollars) of eight employees at a model a good fit? company. The equation y = 0.2x + 38 models the data. Is the model a good fit? Week Sales Y-Value residual x y from Age Salaries y Y- residual Model x Value 1 19 from 2 15 Model 35 42 3 13 37 44 4 11 41 47 5 10 43 50 6 8 45 52 7 7 47 51 8 5 53 49 55 45 Monitoring Progress 1: The table shows the attendance y (in thousands) at an amusement park from 2005 to 2014, when x = 0 represents the year 2005. The equation y = -9.8x + 850 models the data. Is the model a good fit? Year Attendance, Y- residual x y Value from Model 0 850 1 845 2 828 3 798 4 800 5 792 6 785 7 781 8 775 9 760 Example 5: Tell whether a correlation is likely in the situation. If so, tell whether there is a causal relationship. Explain your reasoning. a. time spent exercising and the number of calories Example 4: Use the graph or the equation to: burned. y = 12.0x + 35 b. the number of banks and the population of a city. Monitoring Progress 4: Tell whether a a. approximate the duration before a time of 77 minutes correlation is likely in the situation. If so, tell whether there is a causal relationship. Explain your reasoning. b. predict the time after an eruption lasting 7 minutes a. Playing Video games and grade point average. 4.6 Arithmetic Sequences Notes Vocabulary Definition Picture Sequence Term Arithmetic sequence Common difference Example 1: Write the next three terms of the Monitoring Progress: Write the next three arithmetic sequence. terms of the arithmetic sequence. -7, -14, -21, -28 1. -12, 0, 12, 24, … 2. 0.2, 0.6, 1, 1.4, … 3. 4, 3¾, 3½, 3¼, … Example 2: Graph the arithmetic sequence Monitoring Progress: Graph the arithmetic sequence, what do you notice? 4, 8, 12, 16, … What do you notice? 4. 3, 6, 9, 12, … 5. 4, 2, 0, -2, … 6. 1, 0.8, 0.6, 0.4, … Example 3: Does the graph represent an arithmetic Monitoring Progress 7: Does the graph sequence? Explain? represent an arithmetic sequence? Explain? Example 4: Write an equation for the nth term of Monitoring Progress: Write an equation for the arithmetic sequence 14, 11, 8, 5, … Then find a50. the nth term of the arithmetic sequence. Then find a25. 8. 4, 5, 6, 7, … 9. 8, 16, 24, 32, … 10. 1, 0, -1, -2, … Example 5: Online bidding for a purse increases by Monitoring Progress 11: A carnival charges $5 for each bid after the $60 initial bid. $2 for each game after you pay a $5 entry fee. a. Write a function that represents the arithmetic sequence. b. Graph the function. a. Write a function that represents the arithmetic sequence. b. Graph the function. c. the winning bid is $105 how many bids were there? c. How many games can you play when you take $29 to the carnival? 4.7 Piecewise Functions Notes Vocabulary Definition Picture Piecewise functions Step function Absolute value function Example 1 Evaluate the function f Monitoring Progress: above when a. x = 0 b. x = 4 1. f(-8) 2. f(-2) 3. f(0) Example 2 Graphing a Piecewise 4. f(3) 5. f(5) 6. f(10) Monitoring Progress: Graph the function and Domain: Range: describe the domain & Range Monitoring Progress: Graph the function and describe the domain & Range Example 3 Write a piecewise function for the graph. Monitoring progress: Write a piecewise function for the graph. Example 4: You rent a karaoke machine for 5 days. Monitoring Progress 10: A landscaper rents The rental company charges $50 for the first day and a wood chipper for 4 days. The rental company $25 for each additional day. Write and graph a step charges $100 for the first day and $50 for each function that represents the relationship between the additional day. Write and graph a step function that number x of days and the total cost y (in dollars) of represent the relationship between the number of x days and the total cost y (in dollars) of renting the renting the karaoke machine. chipper. Step 1: use table to organize Info Step 2 Write the step function Step 3: Graph the step function .