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Transcript
H. Algebra 2
1.1 Notes
1.1 (Day One) Domain, Range, and End Behavior
Date: ____________
Learning Target A: I can represent an interval on a number line using an inequality, set notation,
and interval notation.
Interval: a part of a number line without any breaks
Finite Interval: has two endpoints which
may or may not be included in the interval
Finite Interval: is unbounded on one or both
ends
We have different notations we can use to represent intervals on a number line. Fill in the
table below with the given information:
Description of
Interval
All real numbers
from a to b,
including a and b
All real numbers
greater than a
Type of
Interval
Inequality
Set Notation
Interval
Notation
finite
π‘Žβ‰€π‘₯≀𝑏
{π‘₯|π‘Ž ≀ π‘₯ ≀ 𝑏}
[π‘Ž, 𝑏]
infinite
π‘₯>π‘Ž
{π‘₯|π‘₯ > π‘Ž}
(π‘Ž, ∞)
All real numbers
less than or equal
to a
Using what you gathered from above, fill out each table for the given number lines:
A)
Finite
Interval
Inequality
Set Notation
Interval
Notation
B)
Finite
Interval
Inequality
Set Notation
Interval
Notation
1
H. Algebra 2
1.1 Notes
Reflect. Consider the interval shown on the number line.
1. Consider the interval shown on the number line.
a) Represent the interval using interval notation
b) What numbers are included in the interval?
2. What do the intervals [0, 5], [0, 5), π‘Žπ‘›π‘‘ (0, 5) have in common? What makes them different?
3. The symbol βˆͺ represents the union of two sets. What do you think the notation (βˆ’βˆž, 0) βˆͺ (0, ∞)
represents?
Identifying a Function’s Domain, Range, and End Behavior from its Graph
Learning Target B: I can identify a function’s domain, range, and end behavior from its graph.
Given the function 𝒇, define the following:
Gv
Domain of 𝒇: the set of ____________ values, x.
Range of 𝒇: the set of ____________ values, 𝑓(π‘₯).
State the domain and range for the graph of the linear function as an
inequality, using set notation, and using interval notation.
End Behavior: describes what happens to the 𝑓(π‘₯) values as the π‘₯-values either increase
without bound (approach positive infinity) or decrease without bound (approach negative
infinity).
For example, using the graph from above:
As x increases without bound, or as x approaches positive infinity, we see that the 𝑓(π‘₯) values are also
increasing and approaching positive infinity.
This is written:
Describe the end behavior as x decreases without bound, or as x approaches negative infinity.
2
H. Algebra 2
1.1 Notes
For each graph, give the domain and range as an inequality, using set notation, and using
interval notation. Then describe the end behavior.
A) The graph of the quadratic function , 𝑓(π‘₯) = π‘₯ 2 is shown.
B)The graph of the exponential function 𝑓(π‘₯) = 2π‘₯ is shown.
C) The graph of the absolute value function 𝑓(π‘₯) = |π‘₯| + 3 is shown.
D) The graph of a function is shown.
E) 𝑓(π‘₯) = βˆ’π‘₯ 2
3
H. Algebra 2
1.1 Notes
1.1 (Day Two) Domain, Range, and End Behavior
Date: ____________
Review. Graphing Linear Functions
Given each linear function, identify its slope and y-intercept, then draw its graph.
A) 𝑓(π‘₯) = π‘₯ + 1
1
B) 𝑓(π‘₯) = 2 π‘₯ βˆ’ 3
3
C) 𝑓(π‘₯) = βˆ’ 2 π‘₯ + 5
Graphing a Linear Function on a Restricted Domain
Learning Target C: I can graph a linear function on a restricted domain.
Unless otherwise stated, a function is assumed to have a domain consisting of all real numbers for
which the function is defined. Many functionsβ€”such as linear, quadratic and exponential
functionsβ€”are defined for all real numbers, so their domain is all real numbers, which can be
written many different ways.
As the real number set: ____________________
As an Inequality: ___________________________
Set Notation: ________________________________
Interval Notation: __________________________
There are some real-world situations that can be modeled by mathematical functions, but the set of
all real numbers would not make sense for the situation. In these cases, we need to restrict the
domain.
If the rule of a function and its restricted domain are given, you can draw the graph of the function
and then identify the range.
Example 1. For the given function and domain, draw the graph and identify the range using
the same notation as the given domain.
3
A) 𝑓(π‘₯) = 4 π‘₯ + 2 with domain [βˆ’4, 4].
4
H. Algebra 2
1.1 Notes
B) 𝑓(π‘₯) = βˆ’π‘₯ βˆ’ 2 with domain {π‘₯|π‘₯ > βˆ’3}
Reflect.
1. In Part A, how does the graph change if the domain is (-4, 4) instead of [-4, 4]?
2. In Part B, what is the end behavior as x increases without bound? Why can’t you talk about the
end behavior as x decreases without bound?
1
C) 𝑓(π‘₯) = βˆ’ 2 π‘₯ + 2 with domain βˆ’ 6 ≀ π‘₯ < 2
2
D) 𝑓(π‘₯) = 3 π‘₯ βˆ’ 1with domain (βˆ’βˆž, 3]
5
H. Algebra 2
1.1 Notes
Modeling with a Linear Function
Learning Target D: I can model with a linear function.
Recall that when a real-world situation involves a constant rate of change, a linear function is a
reasonable model for the situation. The situation may require restricting the function’s domain.
Example 2. Write a function that models the given situation. Determine a domain from the
situation, graph the function using that domain, and identify the range.
A) Joyce jogs at a rate of 1 mile every 10 minutes for a total of 40 minutes. (Use inequalities for the
domain and range)
Step 1. Identify and define the independent and dependent variables.
Step 3. Find the y-intercept.
Distance (mi)
Step 2. Find the rate of change.
Step 4. Write the function. Identify the domain.
5
4
3
2
1
10
Step 5. Graph the function. Identify the range.
20
30
40
50
Time (min)
B) A candle 6 inches high burns at a rate of 1 inch every 2 hours for
5 hours. (Use interval notation for domain and range.
C) While standing on a moving walkway at an aiport, you are carried forward 25 feet every 15
seconds for one minute. (Use set notation for the domain and range)
6