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Math II Co-teacher Training

Math II Co-teacher Training. Math II Unit 5 (Part 1). Absolute Value Equations and Inequalities. MM2A1c. Solve absolute value equations and inequalities analytically, graphically and by using appropriate technology. EQ #1. Can an absolute value equation ever have “no solution”?.

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Math II Co-teacher Training

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  1. Math II Co-teacher Training Math II Unit 5 (Part 1)

  2. Absolute Value Equations and Inequalities

  3. MM2A1c Solve absolute value equations and inequalities analytically, graphically and by using appropriate technology.

  4. EQ #1 Can an absolute value equation ever have “no solution”?

  5. Absolute Value Recap • Symbol lxl • The distance x is from 0 on the number line. • Always positive • Ex: l-3l= 3 -4 -3 -2 -1 0 1 2

  6. Ex: |x | = 5 • What are the possible values of x? x = 5 or x = -5 • Another example… |x|= 13 • What are the possible values of x?

  7. Now, let’s think about this… • What will happen if …│x + 6│= 16? • What will the two equations be? • What are the solutions? • Another example… • │x │= -8 … What are the solutions? • Remember…Absolute value equations cannot be equal to a negative number!

  8. Solving Absolute Value Equations

  9. To solve an absolute value equation: |ax+b | = c, where c > 0 To solve, set up 2 new equations, then solve each equation. ax+b = c or ax+b = -c ** make sure the absolute value is by itself before you split to solve.

  10. Ex: Solve |6x - 3| = 15 6x - 3 = 15 or 6x - 3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

  11. Ex: Solve |2x + 7| -3 = 8 Get the absolute value part by itself first! |2x+7| = 11 Now split into 2 parts. 2x+7 = 11 or 2x+7 = -11 2x = 4 or 2x = -18 x = 2 or x = -9 Check the solutions.

  12. Revisit EQ #1 Can an absolute value equation ever have “no solution”?

  13. EQ #2 How do absolute value equations and absolute value inequalities differ?

  14. Solving Absolute Value Inequalities

  15. Solving Absolute Value Inequalities • |ax+b| < c, where c > 0 Becomes an “and” problem Changes to: –c < ax+b < c • |ax+b| > c, where c > 0 Becomes an “or” problem Changes to: ax+b > c or ax+b < -c

  16. Solve & graph. • Becomes an “and” problem -3 7 8

  17. Solve & graph. • Get absolute value by itself first. • Becomes an “or” problem. -2 3 4

  18. Revisit EQ #2 How do absolute value equations and absolute value inequalities differ?

  19. Piecewise Functions

  20. Interval Notation

  21. Interval notation is another method for writing domain and range. Symbols you need to know… • Open parentheses ( )- means NOT equal to or does not contain that point or value • Closed parentheses [ ] –mean equal to or contains that point or value • Infinity ∞ - if the graph goes forever to the right (domain) or forever up (range) • Negative Infinity −∞ - If the graph goes forever to the left (domain) or forever down (range) • Union Sign ⋃ - means joined together … this part AND this part Brackets Parentheses Whenever there is a break in the graph, write the interval up to the point. Then write another interval for the section of the graph after that part. Put a union sign between each interval to "join" them together. Use the brackets [ ] if the value is part of the graph or contains that point. Use the open parentheses ( ) if the value is not included in the graph. (i.e. the graph is undefined at that point... there's a hole or asymptote, or a jump)

  22. Find your family Take your card (either the graph or the interval notation) to the person who has your “match.” You are finding your TWIN.

  23. Examples…Give the Domain [-1] U [3, ∞) [0, ∞) [-3] U [-2] U [-1, ∞) (-4, 4]

  24. Definition of a Piecewise Function • Piecewise functions are functions that can be represented by more than one equation (a function made with many “PIECES.” • Piecewise functions do not always have to be line segments. The “pieces” could be pieces of any type of graph. • This type of function is often used to represent real-life problems.

  25. Definition of a Piecewise Function • Piecewise functions are functions that can be represented by more than one equation (a function made with many “PIECES.” • Piecewise functions do not always have to be line segments. The “pieces” could be pieces of any type of graph.

  26. Evaluating Piecewise Functions • Each equation corresponds to a different part of the domain. Find 1. f(-1) 2. f(0) 3. f(5)

  27. Characteristics of a Piecewise graph EQ: How can piecewise functions be described?

  28. Characteristics of Piecewise Graphs • Domain – x-values • Range – y-values • X-intercepts (zeros) – points where graph crosses x-axis • Y-intercept– point where graph crosses y-axis • Intervals of Increase/Decrease/Constant – • read from left to right ALWAYS • give x-values only • write in interval notation • Extrema – • Maximum (highest y-value of function) • Minimum – (lowest y-value of function)

  29. Example… • Give the characteristics of the function. • Domain: • Range: • X-intercepts: • Y-intercepts: • Intervals of increase/decrease/ constant: • Extrema:

  30. Points of discontinuity EQ: How do I identify points of discontinuity?

  31. Continuous Function • Notice that in this case the graph of the piecewise function is one continuous set of points because the individual graphs of each of the three pieces of the function connect. • This is not true of all cases. The graph of a piecewise function may have a break or a gap where the pieces do not meet.

  32. Discontinuous Function • This is not true of all cases. The graph of a piecewise function may have a break or a gap where the pieces do not meet. • The “breaks” or “holes” are called points of discontinuity. • This graph has a point of discontinuity where x = 2.

  33. Parent Functions Review EQ: What are the six Parent Functions from Math I and what are the characteristics of their graph?

  34. Parent Functions Review • Cut and Paste Activity: Have students match parent function properties to its name on graphic organizer.

  35. Parent Functions Review

  36. Graphing Piecewise Functions

  37. Graphing a step Function

  38. Step Function • A step function is an example of a piecewise function. • Let’s graph this example together.

  39. Step Functions • Ceiling Functions • In a ceiling function, all non-integers are rounded up to the nearest integer. • An example of a ceiling function is when a phone service company charges by the number of minutes used and always rounds up to the nearest integer of minutes. • Floor Functions • In a floor function, all non-integers are rounded down to the nearest integer. • The way we usually count our age is an example of a floor function since we round our age down to the nearest year and do not add a year to our age until we have passed our birthday. • The floor function is the same thing as the greatest integer function .

  40. Step functions EQ: How are graphs of step functions used in everyday life?

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