Transcript Lesson 3.4 ppt – Characteristics of Linear Functions

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3.3.1: Identifying Key Features of Linear and Exponential Graphs
The graph below represents Kim’s distance from home
one day as she rode her bike to meet friends and to do a
couple of errands for her mom before returning home.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
1. Use the graph to describe Kim’s journey.
2. What do the horizontal lines on the graph represent?
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
1. Use the graph to describe Kim’s journey.
• Answers will vary. One possible response: Kim rode her
bike to her friend’s house. She stayed at her friend’s
house for a while. Then she left her friend’s house and
rode to a store, which is even farther away from her
house. She stayed at the store for a short time and
bought a couple of items. Kim then headed back toward
her house, stopping once more to take a picture of a
beautiful statue along the way. She then biked the rest
of the way back home.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
2. What do the horizontal lines represent in the graph?
• The horizontal lines represent times when Kim stayed
at one location. Her distance from home did not
change, but time continued to pass.
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3.3.1: Identifying Key Features of Linear and Exponential Graphs
Lesson 3.4 – Characteristics of Linear Functions
Concepts: Characteristics of Linear Functions
EQ: What are the key features of a linear function?
(Standard F.IF.7)
Vocabulary:
Rate of change
Domain/Range
x and y intercepts
Intervals of Increasing/Decreasing
Extrema (Minimum/Maximum)
Key Features of Linear Functions
Domain & Range
Intercepts (x & y)
Increasing/Decreasing
Extrema (Minimum/Maximum)
Rate of Change
Back
Intervals
Identifying Key Features of a
Linear Function
Domain and Range:
Domain: all possible input values
Range: all possible output values
Example: 𝟏, 𝟒 , 𝟐, 𝟓 , (𝟑, 𝟔)
Domain: 1, 2, 3 Range: 4, 5, 6
Identifying Key Features of a
Linear Function
Intercepts:
X-intercept: The place on the x-axis
where the graph
crosses the axis.
-Ordered pair: (x, 0)
Identifying Key Features of a
Linear Function
Intercepts:
X-intercept: The place on the x-axis
where the graph
crosses the axis.
-Ordered pair: (x, 0)
Example 2: y = x + 2
0 = x +2
-2 = x
x-intercept: (-2, 0)
Identifying Key Features of a
Linear Function
Intercepts:
y-intercept: The place on the y-axis
where the graph
crosses the axis
-Ordered pair: (0, y)
Identifying Key Features of a
Linear Function
Intercepts:
y-intercept: The place on the y-axis
where the graph
crosses the axis
-Ordered pair: (0, y)
Example 2: y = x + 2
y = 0 +2
y=2
y-intercept: (0, 2)
Identifying Key Features of a
Linear Function
Increasing or Decreasing????
Increasing: A function is said to increase if
while the values for x increase as well as the
values for y increase. (Both x and y increase)
Identifying Key Features of a
Linear Function
Increasing or Decreasing????
Decreasing: A function is said to decrease if one
of the variables increases while the other
variable decreases. (Ex: x increases, but y
decreases)
Identifying Key Features of a
Linear Function
Intervals:
An interval is a continuous series of values.
(Continuous means “having no breaks”.)
We use two different types of notation for intervals:
1. Brackets ( ) or [ ]
Non-inclusive
2. inequality symbols
≤, ≥, <, >
Inclusive
Ex: [0, 3] and 0< x < 3 both mean all values
between 0 and 3 inclusive
Identifying Key Features of a
Linear Function
Intervals:
A function is positive when its graph is above
the x-axis.
A function is negative when its graph is
below the x-axis.
Identifying Key Features of a Graph
The function is
positive when x > ?
When x ≥ 4!
Or [4, ∞)
Identifying Key Features of a Graph
The function is
negative when x < ?
When x < 4!
Or (-∞, 4)
Identifying Key Features of a
Linear Function
Extrema:
 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 functions will only have a relative
minimum or maximum if the domain is
restricted.
Identifying Key Features of a
Linear Function
Identifying Rate of Change
Rate of Change:
Rate of change or Slope is found by using the
following equation:
y  y
m 
2
1
x 2  x1
Or by reading the rise over the run from a
graph.
Identifying Rate of Change
Identify two points on
the line.
(0, 2) and (5, 1)
Use the formula:
m 
y 2  y1
x 2  x1

1 2
50

1
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Example 1:
Guided
IdentifyPractice
the following:
1. Type of
Example
1 function
and Range
•2.ADomain
taxi company
in Atlanta
3.charges
Y-intercept
$2.75 per ride
4.plus
Intervals
$1.50of
forIncreasing
every mile
or Decreasing
driven.
Determine the key
5.features
Extremaof this function.
6. Rate of Change
Example 2:
Identify the following:
A gear on a machine
1. Type of function
turns at a rate of 3
2. Domain and Range
revolutions per second.
3. Y-intercept
Identify the key features
4. Intervals of Increasing
of the graph of this
or Decreasing
function.
5. Extrema
6. Rate of Change
Example 3:
Identify
An online
the
company
following:
charges
1.
$5.00
Typeaof
month
function
plus $2.00
2.
forDomain
each movie
and Range
you decide
3.
toY-intercept
download.
4. Intervals of Increasing or
Decreasing
5. Extrema
6. Rate of Change
Example 4:
Identify
the following:
A ringtone
company
1. Type
of function
charges
$15 a month
2. Domain
andeach
Range
plus $2 for
3. Y-intercept
ringtone downloaded.
4. Intervals
of Increasing
Create a graph
and then
or
Decreasing
determine
the key
5. Extrema
features of this function.
6. Rate of Change
You Try 1
The starting balance of Adam’s savings account is $575. Each
month, Adam deposits $60.00. Adam wants to keep track
of his deposits so he creates the following equation: f(x) =
60x + 575, where x = number of months.
Identify the following:
1. Type of function
2. Domain and Range
3. Y-intercept
4. Intervals of Increasing or
Decreasing
5. Extrema
6. Rate of Change
You Try 2
Identify
The costthe
of following:
an air conditioner
1.isType
$110.of The
function
cost to run the
2.air
Domain
conditioner
and Range
is $0.35 per
3.minute.
Y-intercept
The table below
4.represents
Intervals of
this
Increasing
relationship.
or
Graph
Decreasing
and identify the key
5.features
Extremaof this function.
6. Rate of Change
3-2-1 Summary
Name 3 new features you learned about today.
Name 2 features you already knew about.
Name 1 feature you still need to practice identifying.