Chapter%201%20Linear%20Functions%20and%20Mathematical%20Modeling%20%20Section%201.4 - PowerPoint PPT Presentation

View by Category
About This Presentation
Title:

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

Description:

Chapter 1 Linear Functions and Mathematical Modeling Section 1.4 – PowerPoint PPT presentation

Number of Views:58
Avg rating:3.0/5.0
Slides: 23
Provided by: Donn2219
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

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


1
Chapter 1 Linear Functions and
Mathematical Modeling Section 1.4

2
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

3
Graphs of Functions
  • The graph of a function f is the set of all
  • points on the plane of the form (x, f(x)).

4
  • 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
5
  • 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

6
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.

7
(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.

8
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

9
(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, ?)

10
  • 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.

11
  • (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.

12
  • 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.

13
  • 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, ?).

14
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.
15
  • 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.

16
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.)  
17
  • 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.

18
  • 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).

19
  • 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.

20
  • (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.

21
  • (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

22
Using your textbook, practice the problems
assigned by your instructor to review the
concepts from Section 1.4.
About PowerShow.com