Lesson 3.6 Key Features of

Exponential Functions

http://youtu.be/FSX1U8u7eb4http://youtu.be/JiaSTKYz2Vg

U.S.S. Arizona Oil-Leakage• On December 6, 1941, the USS Arizona took on a full load

of fuel—nearly 1.5 million gallons—in preparation for its scheduled trip to the mainland later that month.

• The next day, much of it fed the explosion and subsequent fires that destroyed the ship following its attack by Japanese bombers. However, despite the raging fire and ravages of time, some 500,000 gallons are still slowly seeping out of the ship’s submerged wreckage.

• Nearly 70 years after its demise, the USS Arizona continues to spill up to 9 quarts of oil into the harbor each day. The oil is often referred to the “Tears of Arizona” or “Black Tears”.

U.S.S. Arizona Oil-Leakage

Time (years) Annual Oil Leakage (quarts)

1941 4,000,000

1990 3,650

2006 3,468

2009 730

We can take the data from the oil leakageand make a table and a graph from it.

The type of graph that this data will makeis what we call in math an ‘Exponential Decay’.

Concept: Characteristics of Exponential Functions

Lesson EQ: How do we identify the key features of an exponential function? (Standard F.IF.4)

Vocabulary: GrowthDecayAsymptotey-intercept

Lesson 3.6a Key Features of Exponential Functions

Exponential Functions

General form

a = initial value that determines the shapea > 1 stretch; a < 1 shrink; -a = reflection

b = growth if the value is > 1b = decay if the value is between 0 and 1k = horizontal asymptote & vertical shift

Example:

a = _____ Reflection? ______

b = _____ Growth or Decay? _________

Asymptote

Line that a graph approaches but never touches.

Example: k = _____Horizontal asymptote is the line y = _____

y-interceptThe point where the graph crosses the

y-axis. The value of x is 0 at this point.

Example: Substitute 0 for x and solve to find the y-intercept.

y-intercept = _________

Sketch of Graph

Example: • Not a reflection• Growth• Asymptote: y = 0• y-intercept: (0,1)

Example:

a = _____ Reflection? ______

b = _____ Growth or Decay? _________

k = _____ Horizontal Asymptote y = _____

y-intercept ______

Sketch of Graph

Example: • Not a reflection• Decay• Asymptote: y = 0• y-intercept: (0,1)

Example:

a = _____ Reflection? ______

b = _____ Growth or Decay? _________

k = _____ Horizontal Asymptote y = _____

y-intercept ______

Sketch of Graph

Example: • Not a reflection• Growth• Asymptote: y = 1• y-intercept: (0,2)

Example:

a = _____ Reflection? ______

b = _____ Growth or Decay? _________

k = _____ Horizontal Asymptote y = _____

y-intercept ______

Sketch of Graph

Example:

• A reflection• Decay• Asymptote: y = 3• y-intercept: (0,2)

Guided PracticeExample 1Create a table of values for the exponential function f(x) = 2x and graph. State whether it’s a growth or decay and identify the key features.

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3.4.2: Graphing Exponential Functions

Guided Practice: Example 1, continued1. Identify the asymptote of the function.

The asymptote of the function is a line that a graph approaches but never touches.

In the general form the horizontal asymptote is always the constant, k.

In the function f(x) = 2x, the value of k is 0.

The horizontal asymptote of the function is y = 0.

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3.4.2: Graphing Exponential Functions

Guided Practice: Example 1, continued3. Determine the y-intercept of the function.

The y-intercept is where the graph crosses the y-axis. The value of x is 0 at this point.

Substitute 0 for x and solve to find the y-intercept.

y-intercept = (0, 1)

It can also be seen in the table that when x = 0, f(x) =1.

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3.4.2: Graphing Exponential Functions

Guided Practice: Example 1, continued1. Create a table of values.

Choose values of x and solve for the corresponding values of f(x).

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3.4.2: Graphing Exponential Functions

x f(x)–2 –1012

𝑓 (𝑥 )=2𝑥

Growth or Decay?

Guided Practice: Example 1, continued4. Graph the function. Use the table of values to

create a graph of the function.

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3.4.2: Graphing Exponential Functions

x f(x)–2 .25–1 .500 11 22 4

𝑓 (𝑥 )=2𝑥

Guided Practice: Example 1, continued5. State the Domain and Range of the

function.The domain is all x-values. Domain = all real numbers because any number can be used as x.

The range is all y-values.

Range = all numbers > asymptote. y > 0

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3.4.2: Graphing Exponential Functions

✔

Guided Practice: Example 1, continued6. Describe the end behavior of the graph.

The end behavior is what happens at the ends of the graph.

Exponential functions have 2 end behaviors. One towards + or - infinity and the one towards the horizontal asymptote.

As x +∞, y +∞

As x -∞, y 0

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3.4.2: Graphing Exponential Functions

✔

Guided PracticeExample 2Create a table of values for the exponential function and graph. State whether it’s a growth or decay and identify the key features.

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3.4.2: Graphing Exponential Functions

Guided Practice: Example 2, continued1. Create a table of values.

Choose values of x and solve for the corresponding values of f(x).

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3.4.2: Graphing Exponential Functions

x f(x)–2 –1012

Growth or Decay?

Guided Practice: Example 2, continued2. Graph the function. Use the table of values to create a graph of the function.

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3.4.2: Graphing Exponential Functions

x f(x)–2 4–1 20 11 .52 .25

𝑓 (𝑥 )=( 12 )𝑥

Example 2: • Horizontal Asymptote:

• y-intercept:

• Domain: All real #s

• Range:

• End behavior: As x +∞, y

As x -∞, y

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3.4.2: Graphing Exponential Functions

Guided Practice: Example 1, continued

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Think/Pair/Share

• Between you and your partner, who has the longest hair? This person is the 2

• 2s explain to the 1s how they can tell if it is a growth or decay.

• 1s explain to the 2s how to find the y-intercept and asymptote.

• 2s explain to the 1s the domain and range. • 1s explain to the 2s the end behavior.

Your turn……For each problem graph the function and identify the y-intercept, asymptote, domain and range, and end behavior.

1. f(x) = 3x

• Asymptote:

• y-intercept:

• Domain:

• Range:

• End behavior:

2.

• Asymptote:

• y-intercept:

• Domain:

• Range:

• End behavior:

Guided PracticeExample 3Create a table of values for the exponential function 1 and graph. State whether it’s a growth or decay and identify the key features.

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3.4.2: Graphing Exponential Functions

Guided Practice: Example 3, continued1. Create a table of values.

Choose values of x and solve for the corresponding values of f(x).

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3.4.2: Graphing Exponential Functions

x f(x)–2 –1012

Growth or Decay?

Guided Practice: Example 3, continued2. Graph the function. Use the table of values to create a graph of the function.

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3.4.2: Graphing Exponential Functions

x f(x)–2 10/9–1 4/30 21 42 10

𝑓 (𝑥 )=3𝑥+1

Example 3: • Horizontal Asymptote:

• y-intercept:

• Domain: All real #s

• Range:

• End behavior: As x +∞, y

As x -∞, y

Summarizing Strategy: Example for Absent friend

Your absent friend needs you to show them an example of what they missed.Choose 3 of the following 5 features to identify for this exponential function: f(x) = 3x – 2• Asymptote• y-intercept• Domain• Range• End Behavior