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Chapter 1

Chapter 1 . Linear Relations and Functions. Table Of Content Section 1-1 Section 1-2 Section 1-3 Section 1-4 Section 1-5 Section 1-6 Section 1-7 Section 1-8. Section 1-1. Relations and Functions. Relation – A set of ordered pairs. Example: { (1 , 2) , (4 , 7 ) , (3 , 5 ) }

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Chapter 1

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  1. Chapter 1 Linear Relations and Functions

  2. Table Of Content Section 1-1 Section 1-2 Section 1-3 Section 1-4 Section 1-5 Section 1-6 Section 1-7 Section 1-8

  3. Section 1-1 Relations and Functions

  4. Relation – A set of ordered pairs. • Example: { (1 , 2) , (4 , 7) , (3 , 5 ) } • Domain – The first set of numbers in the ordered pair. • Example: x-values {1,3,4} • Range – The second set of numbers in the ordered pair • Example: y-values {2,5,7} List the relations: List the Domain: List the Range:

  5. Example: State each relation as a set of ordered pairs, then state the domain and range. Relation: Domain: Range: { (2,3) , (3,-3) , (4,6) , (6,9) , (7,-2) , (10,8) , (12, -7) , (15,4) , (17,12) , (19,11)} {x | x = 2, 3, 4, 6, 7, 10, 12, 15, 17, 19} {y |y = -7, -3, -2, 3, 6, 8, 9, 11, 12 }

  6. Interval Notation ____________________________ Stating the domain and range of a relation in interval notation Greater (or Less) than and Equal to: -2 ≤ x ≤ 4 Interval Notation [Closed]: [ ] Great than or Less than: -4 < x < 4 Interval Notation (Open): ( ) ____________________________ Infinity Interval Notation: Can a function ever touch infinity? So would it be an open bracket or a closed? x ≥ 4 x ≤ 4 [ 4 , ∞ ) ( -∞ , 4 ] x > 2 x <2 ( 2 , ∞ ) (- ∞ , 2 )

  7. Ex: State the domain and range of each relation in interval notation Domain: 0 ≤ x ≤ 5 [ 0 , 5 ] Range: 50 ≤ x ≤ 100 [ 50 , 100 ] Domain: All Real Numbers (- ∞ , ∞ ) Range: All Real Numbers (- ∞ , ∞ )

  8. Ex: State the domain of each relation in interval notation y = y = y = y =

  9. Determining if the relation is a function Function – A relation in which each element of the domain is paired with EXACTLY ONE element in the range. (meaning… you cannot repeat x values) Vertical Line Test – If a vertical line passes through no more than one point of a relation, then it is a function. Are the following relations functions?

  10. Examples: Determining if the relation is a function

  11. Putting it all together : State the domain and range of each relation, then determine if it is a function.

  12. Function Notation Function Notation: f(x) Say: f of x This is the exact same thing as “y” we are just now using it in terms of functions. Example: y = 2x2 + 3x – 4 is the same as f(x) = 2x2 +3x – 4 Why are we using f(x) then? So we can evaluate functions at a certain x value. Previously – you would solve for y when x would equal a value, for example: y = 2x2 + 3x – 4 , solve when x = 2 Now – you will just evaluate a function, for example: f(2) = 2x2 + 3x – 4

  13. Evaluating Function Evaluate each function for the given value. Graph the function on a calculator, then state the domain and range of the function. f(-2) if f(x) = 2x2 + 5x + 7 g(4) if g(x) = |x – 8|

  14. Homework: Page 10: #4, 6, 9 – 15 A, 32, 33, 38 – 46 A, 51, 53, 57

  15. Pg 10: 4) Keisha is correct. Since a function can be expressed as a set of ordered pairs, a function is always a relation. However, in a function, there is exactly one y-value for each x-value. Not all relations have this constraint. 6) 9) 10) Yes 11) No 12a) 12b) Yes, it passes the vertical line test 13) 84 14) 15)

  16. 32) Yes 33) No 38) No, because it fails the vertical line test. 39) Yes, because each x-value is paired with exactly one y-value. 40) No, because it fails the vertical line test. 41) 9 42) 12 43) 2 44) 45) 46) 51) 53) 57) B

  17. Section 1-2 Composition of Functions

  18. Operations of Functions Sum: Difference: Product: Quotient: (

  19. Examples: Operations of functions 1. Given find each function (f + g)(x) (f – g)(x) c. (f ∙ g)(x) d. (f g)(x)

  20. Compositions of Functions ( • Example: • To Find : • Plug in g(x) into the equation • Then distribute and simplify Try on your own: Find

  21. Ex: State the domain of for Domain for each function: Domain: [4 , ∞) Domain: (∞, 0) U (0, ∞) The domain of the first function has to be true for the second function So…. Therefore… The domain of Because we still have to use the domain restrictions for the 2nd function as well.

  22. Examples of finding domain of Step 1: Find the domain for each Domain of Domain of Step 2: So we need to set the domain for So the domain is (∞, -2) U (-2, ∞)

  23. Iteration Iteration – The composition of a function and itself. So you need the 1st term to find the 2nd, and the 2nd term to find the 3rd. Example: Find the first three iterates: , , of the function f(x) = 2x – 3 for an initial value of = f() = f(1) = 2(1) – 3 = = f() = f(1) = 2(1) – 3 = f() = f() = 2() – 3 =

  24. Homework: Page 17: # 10, 11, 12, 16 – 20 A, 22 – 24 A, 26, 27, 29, 30, 39

  25. Page 17: 10a) 10b) 11) 12) = ; 16) 17)

  26. 18) 19) 20) 26) 27) 29) Yes if f(x) and g(x) are both lines, the can both be represented as f(x)=m1x+b1 and g(x)=m2x+b2. Then when f(g(x)) is applied you get f(g(x)) = m1(m2x+b2)+b1 f(g(x)) = m1m2x + m1b2 + b1 Since each m and b terms are constants, then when they multiply, they will remain a constant. So, f(g(x)) is a linear function if f(x) and g(x) are linear. 22) 23) 24)

  27. 30a) 30b) 2000 J 39) C

  28. Section 1-3 Graphing Linear Equations

  29. Reviewing Linear Equations Linear Equation – Standard Form: Ax + By + C = 0 , where A ≠ 0 and B Slope-Intercept Form: y = mx + b Point slope Form: ) Slope:

  30. Types of Slopes Zeros of a Function Zeros exists when f(x) = 0 Ex: Find the zero of the function.

  31. Graphing Linear Functions Graph using: Point Slope – Graph the given point, then use the slope to continue the graph. Slope Intercept – Graph the intercept, then use the slope to continue the graph. Standard form – Find the x and y intercept, the graph the two points.

  32. Examples Graph the following function: Find the zero of each function. If no zero exists, write none.

  33. Homework: Pg 24: #12, 14, 24, 34 – 37 A, 41, 46

  34. Pg 24: 12) 14) 24) (-5/9, 0) 34) 24 35a) m=1/4 35b)For each 1 degree increase in the temperature, there is a ¼ Pascal increase in pressure. 36) No; the product of two positives is positive, so for the product of the slopes to be -1, one of the slopes must be positive and the other must be negative.

  35. 37a) 36; the software has no monetary value after 36 months. 37b) -290; For ever 1 month change in the number of months, there is a $290 decrease in the value of the software. 41a) d(p)=0.88p 41b) r(d) = d – 100 41c) r(p) = 0.88p – 100 41d) ($799.99, $603.99) ($999.99, $779.99) ($1499.99, $1219.99) 46) D

  36. Section 1-4 Writing Linear Equations

  37. Examples: Write an equation in slope-intercept form for each line described. Slope = 5 ; y-intercept = -2 Passes through A(4,5) ; slope = 6 Passes through A (1, 5) and B( - 8, 9) The y-axis Vertical and passes through (-2,9)

  38. Homework: Pg29: # 12 – 22 E, 23 – 26 A, 29, 35

  39. 24) 25a) 25b) about 5.7 weeks 26) 29) Yes, the slope of the lines through (5,9) and (-3,3) is ¾ . The slope of the lines through (-3,3) and 1/6) is ¾. Since these two lines would have the same slope and would share a point, their equations would be the same. So they are collinear. 35) A Pg 29: 12) 14) 16) 18) 20) 22) 23)

  40. Section 1-5 Writing Equations of Parallel and Perpendicular Lines

  41. Definitions Parallel Lines - Two non-vertical lines in a plane are parallel if and only if their slopes are equal and they have no points in common. Two vertical lines are always parallel. Coincide – Two lines that share the same slope and a point. (The same line) Perpendicular Lines – Two non-vertical lines are perpendicular if and only if their slopes are opposite reciprocals. A horizontal and a vertical line are always perpendicular.

  42. Examples Determine whether the graphs of each pair of equations are parallel, coinciding, perpendicular, or none of these. Write the standard form of the equation of the line that is parallel to the graph of the given equation and passes through the point with the given coordinates. Now do perpendicular

  43. Homework: Pg36: #12-16 E, 23, 26, 28, 31, 40

  44. Pg 36 12) Perpendicular 14) None 16) Coinciding 23) 26) 28a) 28b) 31) 40) 4

  45. Section 1-6 Modeling Real-World Data with Linear Functions

  46. Definitions Best-fit Line – A line that best fits a scatter plot (using a calculator) More referred to as the “regression line” Prediction Equation – The equation that models the data based on what you developed.

  47. Examples The cost of attending college is steadily increasing. To the left, you will see a chart of the average tuition cost. Graph the data. Determine which is the: independent variable (x) and the dependent variable. (y) Create your predicted equation by choosing the two points that best represent the line.

  48. Definitions Goodness of fit: How well a line fits within the data from a scatter plot Correlation coefficient: Describes the Goodness of fit. The closer the data fits to a line, the closer it is to -1 or 1. (Depending on if the data has a positive slope or a negative)

  49. Examples The table contains the fat grams and calories in various McDonald’s Breakfast meals. Use a graphing calculator to find the equation of the regression line and the correlation coefficient. Use the equation to predict the number of calories in a breakfast meal that has 15 grams of fat. Is this accurate? Nutrition facts found at www.mcdonalds.com

  50. Entering Data into the calculator. • To input data into spreadsheet: STAT -> EDIT -> ENTER • In L1 put the Fat grams • In L2 put the Calories • Select Y= -> hit the up arrow and high light PLOT1. • Clear anything out of the y1 columns • Click GRAPH. As you can see nothing is there. • Click ZOOM 9 (which is zoomstat) You should see your data. • Finding the Linear Regression Line. • MODE -> STAT DIAGNOSTICS -> ON • Click STAT -> scroll to CALC at the top -> and hit 4 (which is LinReg(ax+b)) • Leave Xlist and Ylist the same • Leave FreqList blank • For Store RegEQ: (Scroll down so your blinker is on it) • ClickVARS -> scroll to Y-VARS -> 1 (functions) -> 1 (Y1) • Click CALCULATE

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