Transcript Guided Practice: Example 2, continued

Introduction
Real-world contexts that have two variables can be
represented in a table or graphed on a coordinate plane.
There are many characteristics of functions and their
graphs that can provide a great deal of information. These
characteristics can be analyzed and the real-world context
can be better understood.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts
• One of the first characteristics of a graph that we can
observe are the intercepts, where a function crosses
the x-axis and y-axis.
• The y-intercept is the point at which the graph
crosses the y-axis, and is written as (0, y).
• The x-intercept is the point at which the graph
crosses the x-axis, and is written as (x, 0).
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
• Another characteristic of graphs that we can observe
is whether the graph represents a function that is
increasing or decreasing.
• When determining whether intervals are increasing or
decreasing, focus just on the y-values.
• Begin by reading the graph from left to right and
notice what happens to the graphed line. If the line
goes up from left to right, then the function is
increasing. If the line is going down from left to right,
then the function is decreasing.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
• A function is said to be constant if the graphed line is
horizontal, neither rising nor falling.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
•
An interval is a continuous series of values.
(Continuous means “having no breaks.”) A function
is positive on an interval if the y-values are greater
than zero for all x-values in that interval.
•
A function is positive when its graph is above the
x-axis.
•
Begin by looking for the x-intercepts of the function.
•
Write the x-values that are greater than zero using
inequality notation.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
•
A function is negative on an interval if the y-values
are less than zero for all x-values in that interval.
•
The function is negative when its graph is below the
x-axis.
•
Again, look for the x-intercepts of the function.
•
Write the x-values that are less than zero using
inequality notation.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
• Graphs may contain extrema, or minimum or
maximum points.
• A relative minimum is the point that is the lowest, or
the y-value that is the least for a particular interval of a
function.
• A relative maximum is the point that is the highest, or
the y-value that is the greatest for a particular interval
of a function.
• Linear and exponential functions will only have a
relative minimum or maximum if the domain is
restricted.
3.3.1: Identifying Key Features of Linear and Exponential Graphs
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Key Concepts, continued
• The domain of a function is the set of all inputs, or
x-values of a function.
• Compare the following two graphs. The graph on the
left is of the function f(x) = 2x – 8. The graph on the
right is of the same function, but the domain is for
x ≥ 1. The minimum of the function is –6.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
•
Functions that represent real-world scenarios often
include domain restrictions. For example, if we were
to calculate the cost to download a number of ebooks, we would not expect to see negative or
fractional downloads as values for x.
•
There are several ways to classify numbers. The
following slide lists the most commonly used
classifications when defining domains.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
Natural numbers
1, 2, 3, ...
Whole numbers
0, 1, 2, 3, ...
a
..., –3, –2, –1, 0, 1, 2, 3,b...
Integers
Rational numbers
numbers that can be written as
, where a and
b are integers and b ≠ 0; any number that can be written as a
a
b
decimal that ends or repeats
Irrational numbers numbers that cannot be written as
, where a
and b are integers and b ≠ 0; any number that cannot be written as a
decimal that ends or repeats
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Real numbers
the set of all rational and irrational numbers
3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
• An exponential function in the form f(x) = ax, where
a > 0 and a ≠ 1, has an asymptote, or a line that the
graph gets closer and closer to, but never crosses or
touches.
• The function in the following graph has a horizontal
asymptote at y = –4.
• It may appear as though the graphed line touches
y = –4, but it never does.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Key Concepts, continued
•
Fairly accurate representations of functions can be
sketched using the key features we have just described.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Common Errors/Misconceptions
• believing that exponential functions will eventually
touch or intersect an asymptote
• incorrectly identifying the type of function as either
exponential or linear
• misidentifying key features on a graph
• incorrectly choosing the domain for a function
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice
Example 1
A taxi company in Atlanta
charges $2.75 per ride
plus $1.50 for every mile
driven. Determine the key
features of this function.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 1, continued
1. Identify the type of function described.
We can see by the graph that the function is
increasing at a constant rate.
The function is linear.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 1, continued
2. Identify the intercepts of the graphed
function.
The graphed function crosses the y-axis at the point
(0, 2.75).
The y-intercept is (0, 2.75).
The function does not cross the x-axis.
There is not an x-intercept.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 1, continued
3. Determine whether the graphed function
is increasing or decreasing.
Reading the graph left to right, the y-values are
increasing.
The function is increasing.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 1, continued
4. Determine where the function is positive
and negative.
The y-values are positive for all x-values greater
than 0.
The function is positive when x > 0.
The y-values are never negative in this scenario.
The function is never negative.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 1, continued
5. Determine the relative minimum and
maximum of the graphed function.
The lowest y-value of the function is 2.75. This is
shown with the closed dot at the coordinate (0, 2.75).
The relative minimum is 2.75.
The values increase infinitely; therefore, there is no
relative maximum.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 1, continued
6. Identify the domain of the graphed
function.
The lowest x-value is 0 and it increases infinitely.
x can be any real number greater than or equal to 0.
The domain can be written as x ≥ 0.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 1, continued
7. Identify any asymptotes of the graphed
function.
The graphed function is a linear function, not an
exponential; therefore, there are no asymptotes for
this function.
✔
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 1, continued
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice
Example 2
A pendulum swings to 90% of
its height on each swing and
starts at a height of 80 cm.
The height of the pendulum in
centimeters, y, is recorded
after x number of swings.
Determine the key features
of this function.
Number of
swings (x)
Height in
cm (y)
0
80
1
72
2
64.8
3
58.32
5
47.24
10
27.89
20
9.73
40
1.18
60
0.14
80
0.02
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 2, continued
1. Identify the type of function described.
The scenario described here is that of an
exponential function.
We can be certain of this because the pendulum
swings at 90% of its height in each swing; also, we
can see from the table that the values for y do not
decrease at a constant rate.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 2, continued
2. Identify the intercepts of the function
based on the information in the table.
The function crosses the y-axis at the point (0, 80)
as indicated in the table.
The y-intercept is (0, 80).
As the x-values increase, the y-values get closer and
closer to 0, but do not seem to reach 0; therefore,
there is not an x-intercept.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 2, continued
3. Determine whether the function is
increasing or decreasing.
As the x-values increase, the y-values decrease.
The function is decreasing.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 2, continued
4. Determine where the function is positive
and negative.
The y-values are positive for all x-values greater
than 0.
The function is positive when x > 0.
The y-values are never negative in this scenario.
The function is never negative.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 2, continued
5. Determine the relative minimum and
maximum of the function.
The data in the table do not change at a constant
rate; therefore, the function is not linear.
Based on the information given in the problem and
the values in the table, we know that this is an
exponential function.
Exponential functions do not have a relative
minimum because the graph continues to become
infinitely smaller.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 2, continued
The height of the pendulum never goes higher than
its initial height; therefore, the relative maximum of
this function is (0, 80).
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 2, continued
6. Identify the domain of the function.
The lowest x-value is 0 and it increases infinitely.
x can be any real number greater than or equal to 0,
but cannot be a partial swing.
The domain is all whole numbers.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 2, continued
7. Identify any asymptotes of the function.
The points approach 0, but never actually reach 0.
The asymptote of this function is y = 0.
✔
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Guided Practice: Example 2, continued
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3.3.1: Identifying Key Features of Linear and Exponential Graphs