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Thinker….

Thinker…. Lew is playing darts on a star-shaped dartboard in which two equilateral triangles trisect the sides of each other as shown. Assuming that a dart hits the board, what is the probability that it will land inside the hexagon?. Solution to the “Thinker”. 1/2.

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Thinker….

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  1. Thinker…. Lew is playing darts on a star-shaped dartboard in which two equilateral triangles trisect the sides of each other as shown. Assuming that a dart hits the board, what is the probability that it will land inside the hexagon?

  2. Solution to the “Thinker” 1/2. As shown, each triangle can be reflected to the interior of the hexagon in such a way that the triangle areas are equal to the area of the hexagon. In this manner, the area of the hexagon is half that of the entire dartboard.

  3. FunctionsDomain & RangeIncreasing & Decreasing

  4. What is a relation? A set of ordered pairs. (x, y) Could you represent a relation another way?

  5. Domain vs. Range • Domain- the set of x-coordinates of a relation or function. • Range- the set of y-coordinates of a relation or function. Notice anything about the order of the domain/range?

  6. Functions • Discrete- a graph which consists of points which are not connected.

  7. Continuous Data What do you think of when you hear the word continuous? A function that is traceable! Examples: Lines…Parabolas…any others?

  8. Discontinuous Data What do you think of when you hear the word discontinuous? A function that is not traceable. (must pick up your pencil) Would discrete data be continuous or discontinuous?

  9. Piecewise Functions -a graph which consists of line segments or pieces of other nonlinear graphs. Would piecewise functions be continuous or discontinuous? Would piecewise functions be discrete?

  10. What is a function? A function is a relation in which each element of the domain is paired with exactlyone element from the range (no duplicate x-values). A function is denoted as f(x) or pronounced “f of x”.

  11. Some relations are functions…some are not… But…how do we know? Let’s find out!

  12. Draw these two graphs. • Vertical Line Test! • Touches it once, it IS a function! • Touches >1, it is NOT a function!

  13. That’s too hard to remember, can I just graph the points and use the VLT? What if it’s not a graph?? State weather the following relation is a function or not. • Do the x-values repeat? • No they don’t….YES it is a function! • Yes they repeat…NO it’s not a function!

  14. Example 1 • Determine whether the relation {(-1, ), (-1, ), (0, 1)} is a function. • Justify your decision in a complete sentence. This relation is not a function since two values of -1 will not pass the vertical line test.

  15. Examples • State whether each relation or graph below is a function: • {(1,2),(2,4),(3,5)(0,5)} • {(0,4),(2,4),(1,3),(2,5)} • {(&,*),($,%),(#,^),(@,*),(#,@)} Yes No No

  16. no yes no Examples • State whether each relation or graph below is a function:

  17. (5,5) is open yes no Examples • State whether each relation or graph below is a function:

  18. Parenthesis-NOT inclusive is always open Open Interval • An open intervalis the set of all real numbers that lie strictly between two fixed numbers a and b. (a, b) a< x <b ***think about open dots on a number line!

  19. Closed Interval • A closed intervalis the set of all real numbers that lie in between and contain both endpoints a and b. [a, b] a< x <b Brackets mean-inclusive **think about closed dots on a number line!

  20. Half-Open Intervals • Half-open intervals are intervals that contain one but not both endpoints a and b. (a, b] a< x <b [a, b) a< x <b

  21. What do we use this for? • **Intervals will be used to define the domain and range of given functions or graphs which are continuous and/or increasing and decreasing intervals. OTHER SYMBOLS TO KNOW: U : union symbol used to join more than one interval together. 0 (zero): neither positive nor negation.

  22. Number Line Examples Fill in the missing parts in the chart below.

  23. Graph Example 2 Determine the domain, range, and continuity of the graph below.

  24. Graph Example 3 Determine the domain, range, and continuity of the graph below.

  25. Graph Example 4 Determine the domain, range, and continuity of the graph below.

  26. Graph Example 5 Determine the domain, range, and continuity of the graph below.

  27. Increasing/Decreasing • Increasing intervals occur when. . .reading a graph left to right, the interval in which the function is rising. Decreasing intervals occur when. . .reading a graph left to right, the interval in which the function is falling.

  28. Increasing/Decreasing Example 6 a. Determine the interval(s) of x in which f(x) in increasing. (between what two x values is the function increasing?)

  29. Increasing/Decreasing Example 6 b. Determine the interval(s) of x in which f(x) in decreasing. (between what two x values is the function decreasing?)

  30. Increasing/Decreasing Example 6 c. Determine the interval(s) of x in which f(x) is positive. (between what two x values are the y-values positive?)

  31. Reflection • Write a question in your notebook about something Mrs. Gromesh taught today that you aren’t 100% on understanding yet.

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