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Linear Functions

Linear Functions . Solving Linear Equations 1.1 Using Data to create Scatterplots 1.2. Solving Linear Equations 1.1 . Solving for an unknown variable: 5 x + 25 = 40. Solving Linear Equations 1.1.

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Linear Functions

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  1. Linear Functions Solving Linear Equations 1.1 Using Data to create Scatterplots 1.2

  2. Solving Linear Equations 1.1 Solving for an unknown variable: 5x + 25 = 40

  3. Solving Linear Equations 1.1 U-Haul charges $19.95 for the day and $0.79 per mile driven to rent a 10-foot truck. The total cost to rent a 10-foot truck for the day can be represented by the equation U = 19.95 + 0.79m Where U is the total cost in dollars to rent a 10 foot truck from U-Haul for the day and m is the number of miles the truck is driven. • Determine how much it will cost you to rent a 10-foot truck from U-Haul and drive it 75 miles • Determine the number of miles you can travel for a total cost of $175.00.

  4. Solving Linear Equations 1.1

  5. Revenue cost and profit (1.1) A local chiropractor has a small office where she cares for patients. She has $8000 in fixed costs each month that cover her rent, basic salaries, equipment, and utilities. For each patient she sees, she has an average additional cost of about $15. The chiropractor charges her patients or their insurance company $80 for a visit.

  6. Chiropractor 1.1 • Write an equation for the total monthly cost when v patient visits are done in a month. • What is the total monthly cost if the chiropractor has 100 patients visit during a month? • Write an equation for the monthly revenue when v patients visit a month. • Write an equation for the monthly profit the chiropractor makes if she has v patient visits in a month. • What is the monthly profit when 150 patients visit in a month? • How many patient visits does this chiropractor need to have in a month for her profit to be $5000?

  7. Solving Linear Equations 1.1 Solving equations with fractions 2/3 x + 5/6 = 7 • ¼ x + 3/8 = 2 • 2/5(x + 4) = 7/10 x - 5

  8. Solving Linear Functions (1.1) Literal equations: rewrite the equation in terms of the desired variable. Ex: D = rt Solve: H = 1/3 m2r ; for r Solve d = ax + c ; for x

  9. Using Data to create scatterplots1.2 Data We gather data to predict, understand, and study events or cases. Government collects population info Effects of caffeine on health and function Grade distribution on schools

  10. Tables 1.2Organizes and presents the data into a useful form. Quantity of students who answered correctly Questions on an exam.

  11. Graph Visual representation of the information (1.2) Define the variables Independent variable Dependent variable Examples Renting a U-Haul truck and driving it m miles for a day The numbers of births at a local hospital on a monthly basis.

  12. Scatterplots (1.2) Dependent Independent

  13. Create a Scatterplot 1.2 The percentage of adults aged 20 years and over in the United States who are considered obese is given in the table.

  14. Determining Scale (1.2) Scale Spacing for each axis

  15. Redefine the scale (1.2) Adjust definition of variables and data Ex: t = time in years since 2000 Adjustment to the scale makes the graph accurate and not distorted. Once the variables are defined stay consistent with the values used for each variable.

  16. Redefine the scale (1.2) 4 years since 2000

  17. Line of best fit (1.2) Make a prediction based on the line of best fit for the percentage of adults in the US that were considered obese in 2010

  18. Intercepts (1.2) Horizontal intercept (x-intercept) Vertical intercept (y-intercept)

  19. Domain and Range (1.2) Domain = set of values for an independent variable that result in a reasonable output values with no model breakdown What are the x values in the given app problem Range = set of dependent values resulting from given domain values. What is the y values in the app problem

  20. Percentage of students in twelfth grade who report smoking daily is given in the table. (1.2)

  21. Understanding the graph 1.2

  22. Fundamentals of Graphing (1.3) Methods of graphing an equation A table of values (x-y chart) Slope intercept form (y = mx + b) x and y intercepts

  23. Fundamentals of Graphing (1.3) Given an equation: y = 3x – 8 can we graph it on the coordinate plane? x-y table

  24. Graphing on the TI-84 (1.3) On the window plug in the equation y = x Look a) Then graph y = 2x and y= 8x b) Graph y = -x and y = -2x c) Graph y = ½ x and y = 1/8x

  25. Slope intercept form y = mx + b (1.3) What value represents slope? What value represents y-intercept? How can we utilize the info to graph the line?

  26. Slope (1.3) Rise Run Rate of change of the line How much the output variable changes for a unit change in the input variable.

  27. Graphing: what is slope in a word problem? (1.3) Given an equation C = 38m + 150 where C represents the cost in dollars for pest management for a pest control company when m months of service is provided.

  28. Slope (1.3) Example: Let C = 4.5p + 1200 be the total cost in dollars to produce p pizzas a day at a local pizzeria. Example 2: Let D = 0.28t + 5.95 be the percentage of adults aged 18 years old and over in the United states that have been diagnosed with diabetes, t years since 2000. What does the slope mean?

  29. Intercepts (1.4) Intercepts are points where the line crosses the x-axis and the y-axis. Example In the U-Haul rental the vertical intercept (0, 19.95) If rented the truck but did not drive the truck it would cost $19.95 for the day.

  30. Understanding intercepts in application problem (1.4) The number of students who are enrolled in math classes at a local college can be represented by E = -17w +600 Where E represents the math class enrollment at the college w weeks after the start of the fall semester.

  31. Practice (1.4) Find the vertical and horizontal intercepts and explain their meaning in the given situation. Let D = 0.28t + 5.95 be the percentage of adults aged 18 years old and over in the United states that have been diagnosed with diabetes, t years since 2000.

  32. Forms of a line (1.4) Slope-intercept form y = mx +b Standard/General form Ax + By = C A is nonnegative A, B, and C are integers

  33. General Form (1.4) Ax + By = C Rewrite the following equations in general form: y = 3x – 15 y = 1/5x + 3/8

  34. Unique Lines (1.4) Horizontal Lines y = k m = 0 Vertical lines x = k m = undefined

  35. Sketch the graph of the following lines x = -4 y = - 1.5

  36. Finding Equations of Lines (1.5) To write the equation of a line need a) b) Slope-Intercept form Point-Slope form Parallel/Perpendicular lines and a point

  37. Given the following info find the linear equation (1.5) • m = 4 and y-int = -7 • M = -1/2 and y-int = 2 • M= 0 and y-int = 4 • M = undefined x-int = 5

  38. y = mx + b (1.5) Write the equation of a line that passes through the points (4,3) and (20, -17). Determine the slope Utilize one point to find the intercept Substitute the slope and intercept into the slope intercept form. Check the equation by substituting the points that were given.

  39. Straight line depreciation (1.5) A business purchased a production machine in 2005 for $185,000. For tax purposes, the value of the machine in 2011 was $129,500. If the business is using straight line depreciation, write the equation of the line that gives the value of the machine based on the age of the machine in years.

  40. AIDs cases for adolescents (1.5) According to www.childtrendsdatabank.org the number of newly diagnosed AIDS cases for adolescents 13-19 years old in the United States was 310 in 2000 and 458 in 2003. Assume that the number of cases is growing at a constant rate, and write an equation to represent this situation.

  41. Point-Slope form (1.5) (y – y1) = m(x – x1) Write the equation of the line that passes through the points (6,-13) and (18,-31)

  42. Parallel lines (1.5) Never intersect Same direction Same slope

  43. Perpendicular Lines (1.5) Cross at right angles Slope is opposite reciprocals

  44. Line Equation (1.5) Write the equation of the line that goes through the point (-12,8) and is perpendicular to the line y = 4x –23 Write the equation of the line that goes through the point (8,11) and is parallel to the line 5x – 2y = 30

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