thinker…. lew is playing darts on a star-shaped dartboard in which two equilateral triangles...
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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?
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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.
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Functions
Domain & Range
Increasing & Decreasing
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What is a relation?
A set of ordered pairs. (x, y)
Could you represent a relation another way?
)}1,3(),1,0(),2,2(),2,2{(
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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.
}3,0,2{: D
}2,1,2{: RNotice anything about the order of the domain/range?
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Functions• Discrete- a graph which consists of
points which are not connected.
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Continuous DataWhat do you think of when you hear the word continuous?
A function that is traceable! Examples: Lines…
Parabolas…any others?
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Discontinuous DataWhat 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?
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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?
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What is a function?A function is a relation in which each element of the domain is paired with exactly one element from the range (no duplicate x-values). A function is denoted as f(x) or pronounced “f of x”.
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Some relations are functions…some are not…
But…how do we know? Let’s find out!
)}8,4(),3,3(),0,3(),1,9{( f(x)=x
-3 -2 -1 1 2
-3
-2
-1
1
2
x
y
f(x)=-x̂ 2 + 5
-8 -6 -4 -2 2 4 6 8
-6
-4
-2
2
4
6
x
y
f(x)=sin(x*10.5/10)
-3 -2 -1 1 2 3
-2
-1
1
2
x
y
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Draw these two graphs.
• Vertical Line Test!• Touches it once, it IS a function!• Touches >1, it is NOT a function!
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What if it’s not a graph??
)}8,4(),3,3(),0,3(),1,9{(
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!
That’s too hard to remember, can I just graph the points and use the VLT?
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Example 1
• Determine whether the relation {(-1, ), (-1, ), (0, 1)} is a function. • Justify your decision in a completesentence.
This relation is not a function since two values of -1 will not pass the vertical line test.
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Examples
• State whether each relation or graph below is a function:
1. {(1,2),(2,4),(3,5)(0,5)}
2. {(0,4),(2,4),(1,3),(2,5)}
3. {(&,*),($,%),(#,^),(@,*),(#,@)}
Yes
No
No
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Examples•State whether each relation or graph below is a function:
no yes no
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Examples•State whether each relation or graph below is a function:
(5,5) is open
yes no
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Open Interval
• An open interval is the set of all real numbers that lie strictly between two fixed numbers a and b.
(a, b)
a< x <b
Parenthesis-NOT inclusive
is always open
***think about open dots on a number line!
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Closed Interval• A closed interval is the set of all
real numbers that lie in between and contain both endpoints a and b.
[a, b]
a< x <bBrackets mean-inclusive
**think about closed dots on a number line!
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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
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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.
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Number Line Examples
InequalityInterval Notation Graph
-1< x <4
x< 1 or x > 5
Fill in the missing parts in the chart below.
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Graph Example 2
Determine the domain, range, and continuity of the graph below.
x
y
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Graph Example 3
Determine the domain, range, and continuity of the graph below.
x
y
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Graph Example 4
Determine the domain, range, and continuity of the graph below.
x
y
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Graph Example 5
Determine the domain, range, and continuity of the graph below.
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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.
x
y
x
y
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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?)
x
y
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Increasing/Decreasing Example 6
x
yb. Determine the interval(s) of x in which f(x) in decreasing. (between what two x values is the function decreasing?)
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c. Determine the interval(s) of x in which f(x) is positive. (between what two x values are the y-values positive?)
Increasing/Decreasing Example 6
x
y
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Reflection
•Write a question in your notebook about something Mrs. Gromesh taught today that you aren’t 100% on understanding yet.