Chapter%201%20Linear%20Functions%20and%20Mathematical%20Modeling%20%20Section%201.4

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Chapter 1 Linear Functions and
Mathematical Modeling Section 1.4

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Section 1.4 Interpreting and Evaluating
Functions Using Graphs

• Graphs of Functions
• Identifying Domain and Range from Graphs of
  Functions
• Vertical Line Test
• Increasing, Decreasing, and Constant Functions

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Graphs of Functions

• The graph of a function f is the set of all
• points on the plane of the form (x, f(x)).

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• Example
• Construct the graph of f(x) 3x 4
• We can create a table of values, choosing several
  values
• for the input and calculating the corresponding
  outputs.
•  
• Plotting and connecting the points, the graph of
  the given
• function follows

x 1 0 2 3
f(x) 7 4 2 5
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• The graph of a function is shown below.
• a. Find C(100)
• C(100) means given the input (x-value) of 100,
  find the output.
• Therefore, C(100) 15
• b. Find x if C(x) 10
• If C(x) 10, we are looking for x-values where
  the output is10.
• Therefore, x 50 300

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The graph of the function f(x) x2 x 5
is shown below. Use the graph to find the
following. 

• a. Find f(2).
• We locate the output, y, when the input, x, is
  2. f(2) 3,
• which is equivalent to the point (2, 3) on the
  graph.
• b. Estimate and interpret f(0).
• f(0) 5. This is the vertical or y-intercept of
  the function.

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(Contd.) The graph of the function f(x)
x2 x 5 is shown below. Use the graph to find
the following. 

• c. Find the input(s) when the output is 3.
• We look for any point(s) on the graph where y has
  a value of
• 3. Observe that f(2) 3 and f(1) 3. The
  inputs are 2
• and 1.
• d. Find the value(s) of x for which f(x) 1.
• f(x) 1 is equivalent to y 1. Thus, f(x) 1
  for x 3 and
• x 2.

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The graph of the function f(x) x2 x
6 is shown below. Use the graph to answer the
following. a. For what x-values is f(x) ? 4?
Write your answer using interval notation. 

• f(x) ? 4 means the graph lies on or above y 4.
• This occurs when 1 ? x ? 2.
• In interval notation 1, 2

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(Contd.)The graph of the function f(x)
x2 x 6 is shown below. Use the graph to
answer the following. b. For what x-values
is f(x) lt 0? Write your answer using interval
notation. 

• f(x) lt 0 means the graph lies below y 0, that
  is, below the
• x-axis.
• This occurs when x lt 2 and x gt 3.
• In interval notation (?, 2) U (3, ?)

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• The graph below illustrates the number of dogs in
  an animal shelter after a tropical storm has
  passed through a small
• town. Let t represent time in days, and n
  represent number
• of dogs.
• a. Identify the input.
• The input is t, time in days.
• b. Identify the output.
• The output is n, number of dogs.

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• (Contd.)
• c. How many dogs are in the shelter in 3 days?
• 24 dogs
• d. Find n(0) and interpret its meaning.
• n(0) 12. There were 12 dogs in the shelter
  before the
• tropical storm.
• e. Find the value of t for which n(t) 36.
  Interpret this value.
• t 6. Six days after the storm, there were 36
  dogs in the shelter.

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• Estimate the domain and range for the function
  whose graph is shown below.
• Domain (possible values of the input) From the
  smallest
• (leftmost) to the largest (rightmost).
• The domain consists of all x-values less than or
  equal to 8, thus
• the domain is given by the interval (?, 8.
• Range (possible values of the output) From the
  lowest
• (bottommost) to the highest (uppermost).
• The bottommost y-value is 6 and the uppermost
  y-coordinate
• is 6, therefore, the range is given by the
  interval 6, 6.

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• Estimate the domain and range for the function
  whose graph is shown below.
• Domain
• The domain consists of all x-values greater than
  4, thus the
• domain is given by the interval (4, ?).
• Range
• The range consists of all y-values greater than
  or equal to 6,
• thus the range is given by the interval 6, ?).

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Vertical Line Test If a
vertical line meets a graph more than once, the
graph does not represent a function. For each
x-value, there can only be one y-value.
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• True or False The following graph represents a
  function.
• True It passes the Vertical Line Test. Any
  vertical line will intersect this graph only
  once.
• Notice that for each input there is only one
  output.

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Increasing, Decreasing, and Constant Functions
A function is increasing on an interval if it is
rising as it goes left to right on the interval.
(As x-coordinates increase, y-coordinates
increase.) A function is decreasing on an
interval if it is falling as it goes left to
right on the interval. (As x-coordinates
increase, y-coordinates decrease.) A function
is constant on an interval if the graph is a
horizontal line on the interval. (As
x-coordinates increase, y-coordinates remain
constant.)  
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• Notes
• ? We read a graph from left to right, and the
  interval over
• which a function is increasing, decreasing, or
  constant is
• given only in terms of the x-values. So, we use
  interval
• notation referring to the x-coordinates only.
• ? Use open intervals (parentheses, not brackets)
  in the
• interval notation, since the turning/ending
  points are
• neither increasing nor decreasing.

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• Given the graph of y f(x), determine the
  intervals for which the function is (a)
  increasing, (b) decreasing, or (c) constant.
• a. The graph rises (increases) from left to
  right on the
• interval (8, 4).
• b. The graph falls (decreases) from left to
  right on the
• interval (4, 12).
• c. The function is constant over the interval
  (?, 8).

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• The graph below shows the path of a model rocket
  launched upward from the ground at an initial
  velocity of 149 feet per
• second. Its height, h, at t seconds, can be
  modeled by
• h(t) -16t2 149t.
• a. Use the graph to evaluate h(3) and explain its
  meaning in the
• context of the problem.
• h(3) 300.
• After 3 seconds, the model rocket's height going
  up is 300 feet.

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• (Contd.)
• The graph below shows the path of a model rocket
  launched upward from the ground at an initial
  velocity of 149 feet per
• second. Its height, h, at t seconds, can be
  modeled by
• h(t) -16t2 149t.
• b. Estimate the location of the ordered pair
  associated with
• h(4.7) 346.9, and explain its meaning in terms
  of the problem.
• (4.7, 346.9). After 4.7 seconds, the model rocket
  reaches a maximum height of approximately 346.9
  feet.

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• (Contd.)
• c. Estimate the increasing and decreasing
  intervals.
• Increasing (0, 4.7)
• Decreasing (4.7, 9.3)
• d. Estimate the domain and range from the graph.
• Domain 0, 9.3
• Range 0, 346.9

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Using your textbook, practice the problems
assigned by your instructor to review the
concepts from Section 1.4.