1. Lesson 3.6 Key Features of Exponential Functions

2. 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”.

3. U.S.S. Arizona Oil-Leakage
We can take the data from the oil leakage
and make a table and a graph from it.
The type of graph that this data will make
is what we call in math an ‘Exponential Decay’.
Time (years)
Annual Oil Leakage (quarts)
1941
4,000,000
1990
3,650
2006
3,468
2009
730

4. Lesson 3.6a Key Features of Exponential Functions
Concept: Characteristics of Exponential Functions
Lesson EQ: How do we identify the key features of an exponential function? (Standard F.IF.4)
Vocabulary:
Growth
Decay
Asymptote
y-intercept

5. Exponential Functions
General form 𝒇 𝒙 =𝒂 𝒃 𝒙 +𝒌 a = initial value that determines the shape a > 1 stretch; a < 1 shrink; -a = reflection b = growth if the value is > 1 b = decay if the value is between 0 and 1 k = horizontal asymptote & vertical shift

6. Example: 𝒇 𝒙 = 𝟐 𝒙 a = _____ Reflection
Example: 𝒇 𝒙 = 𝟐 𝒙 a = _____ Reflection? ______ b = _____ Growth or Decay? _________

7. Asymptote
Line that a graph approaches but never touches. Example: 𝒇 𝒙 = 𝟐 𝒙 k = _____ Horizontal asymptote is the line y = _____

8. y-intercept
The point where the graph crosses the y-axis. The value of x is 0 at this point. Example: 𝒇 𝒙 = 𝟐 𝒙 Substitute 0 for x and solve to find the y-intercept. y-intercept = _________

9. Sketch of Graph Example: 𝒇 𝒙 = 𝟐 𝒙 Not a reflection Growth
Asymptote: y = 0
y-intercept: (0,1)

10. Example: 𝒇 𝒙 = 𝟏 𝟐 𝒙 a = _____ Reflection
Example: 𝒇 𝒙 = 𝟏 𝟐 𝒙 a = _____ Reflection? ______ b = _____ Growth or Decay? _________ k = _____ Horizontal Asymptote y = _____ y-intercept ______

11. Sketch of Graph Example: 𝒇 𝒙 = 𝟏 𝟐 𝒙 Not a reflection Decay
Asymptote: y = 0
y-intercept: (0,1)

12. Example: 𝒇 𝒙 = 𝟑 𝒙 +𝟏 a = _____ Reflection
Example: 𝒇 𝒙 = 𝟑 𝒙 +𝟏 a = _____ Reflection? ______ b = _____ Growth or Decay? _________ k = _____ Horizontal Asymptote y = _____ y-intercept ______

13. Sketch of Graph Example: 𝒇 𝒙 = 𝟑 𝒙 +𝟏 Not a reflection Growth
Asymptote: y = 1
y-intercept: (0,2)

14. Example: 𝒇 𝒙 = −𝟏(𝟐) 𝒙 +𝟑 a = _____ Reflection
Example: 𝒇 𝒙 = −𝟏(𝟐) 𝒙 +𝟑 a = _____ Reflection? ______ b = _____ Growth or Decay? _________ k = _____ Horizontal Asymptote y = _____ y-intercept ______

15. Sketch of Graph Example: 𝒇 𝒙 = −𝟏(𝟐) 𝒙 +𝟑 A reflection Decay
𝒇 𝒙 = −𝟏(𝟐) 𝒙 +𝟑
A reflection
Decay
Asymptote: y = 3
y-intercept: (0,2)

16. Guided Practice Example 1
Create 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.
3.4.2: Graphing Exponential Functions

17. Guided Practice: Example 1, continued
1. 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 𝑓 𝑥 =𝑎 𝑏 𝑥 +𝑘 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.
3.4.2: Graphing Exponential Functions

18. Guided Practice: Example 1, continued
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.
3.4.2: Graphing Exponential Functions

19. Guided Practice: Example 1, continued Create a table of values.
Choose values of x and solve for the corresponding values of f(x).
𝑓 𝑥 = 2 𝑥
Growth or Decay? x
f(x) –2 –1 1 2
3.4.2: Graphing Exponential Functions

20. Guided Practice: Example 1, continued
Graph the function. Use the table of values to create a graph of the function.
𝑓 𝑥 = 2 𝑥 x
f(x) –2
.25 –1
.50 1 2 4
3.4.2: Graphing Exponential Functions

21. ✔ Guided Practice: Example 1, continued
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 ✔
3.4.2: Graphing Exponential Functions

22. ✔ Guided Practice: Example 1, continued
6. 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 ✔
3.4.2: Graphing Exponential Functions

23. Guided Practice Example 2
Create a table of values for the exponential function 𝑓 𝑥 = 𝑥 and graph. State whether it’s a growth or decay and identify the key features.
3.4.2: Graphing Exponential Functions

24. Guided Practice: Example 2, continued Create a table of values.
Choose values of x and solve for the corresponding values of f(x).
𝑓 𝑥 = 𝑥
Growth or Decay? x
f(x) –2 –1 1 2
3.4.2: Graphing Exponential Functions

25. Guided Practice: Example 2, continued
2. Graph the function. Use the table of values to create a graph of the function.
𝑓 𝑥 = 𝑥 x
f(x) –2 4 –1 2 1 .5
.25
3.4.2: Graphing Exponential Functions

26. Example 2: 𝑓(𝑥) = 1 2 𝑥 Horizontal Asymptote: y-intercept:
Domain: All real #s
Range:
End behavior:
As x +∞, y
As x -∞, y

27. Guided Practice: Example 1, continued 27
3.4.2: Graphing Exponential Functions

28. 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.

29. Your turn……
For each problem graph the function and identify the y-intercept, asymptote, domain and range, and end behavior.
𝑓(𝑥) =3𝑥
𝑓(𝑥) = 𝑥

30. 1. f(x) = 3x
Asymptote:
y-intercept:
Domain:
Range:
End behavior:

31. 2. 𝑓(𝑥) = 𝑥
Asymptote:
y-intercept:
Domain:
Range:
End behavior:

32. Guided Practice Example 3
Create a table of values for the exponential function 𝑓 𝑥 = 3 𝑥 +1 and graph. State whether it’s a growth or decay and identify the key features.
3.4.2: Graphing Exponential Functions

33. Guided Practice: Example 3, continued Create a table of values.
Choose values of x and solve for the corresponding values of f(x).
𝑓 𝑥 = 3 𝑥 +1
Growth or Decay? x
f(x) –2 –1 1 2
3.4.2: Graphing Exponential Functions

34. Guided Practice: Example 3, continued
2. Graph the function. Use the table of values to create a graph of the function.
𝑓 𝑥 = 3 𝑥 +1 x
f(x) –2
10/9 –1
4/3 2 1 4 10
3.4.2: Graphing Exponential Functions

35. Example 3: 𝑓 𝑥 = 3 𝑥 +1 Horizontal Asymptote: y-intercept:
Domain: All real #s
Range:
End behavior:
As x +∞, y
As x -∞, y

36. 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