Algebra II
1.1 Domain, Range, and End Behavior
1.1 Domain, Range, and End Behavior
Relate the domain of funcons to its graph, and where applicable, to the quantave relaonship it describes.
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Algebra II
1.1 Domain, Range, and End Behavior
Essential Question: How can you determine the domain, range, and end behavior of a function?
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
( ­ ∞, +∞)
all real numbers
All three intervals contain all the numbers between 0 and 5. The interval [0, 5] also includes the endpoints 0 and 5, the interval [0, 5) includes only the endpoint 0, and the interval (0, 5) does not contain either endpoint.
All real numbers except 0.
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
Unlike the graph of a linear function, the graph of a quadratic function has a turning point (the vertex), which changes the direction of the graph
The f(x)­values do not decrease without bound. They instead approach 0.
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
The graph no longer includes the endpoints of the segment.
As x +∞, f(x) ­∞. Because the domain does not include values of x that are less than or equal to ­3, the values of x cannot decrease without bound. 15
Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
The domain is 0 < t < 30 instead of 0 < t < 40
The graph's right endpoint is (30, 3) instead of (40, 4)
The range is 0 < d < 3 instead of 0 < d < 4
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Algebra II
1.1 Domain, Range, and End Behavior
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Algebra II
1.1 Domain, Range, and End Behavior
[a, b]: This interval includes all real numbers between a and b, including both a and b.
[a, b): This interval includes all real numbers between a and b, including a but not b.
(a, b]: This interval includes all real numbers between a and b, including b but not a.
(a, b): This interval includes all real numbers between a and b but does not include a or b.
If the domain is bounded at both ends, the graph of the linear function is a line segment rather than a line. If the domain is bounded at only one end, the graph of the linear function is a ray rather than a line.
If slope is positive, the f(x)­values increase without bound as the x­
values increase without bound, and the f(x)­values decrease without bound as the x­values decrease without bound. If the slope is negative, the end behavior reverses: The f(x)­values decrease without bound as the x­values increase without bound, and the f(x)­values increase without bound as the x­values decrease without bound.
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