1. 7.1 & 7.2 - Graphing Exponential Transformations
Grade Distribution
1st
4th
7th
9th A 15 8
n/a B 10 C 2 3 D F
No Shows
100+ 1
Range
78-105
47-101
Avg
89.72
83.5
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7.1 & Graphing Exponential Transformations

2. Graphing Exponential Functions
Section 7.1 and 7.2,
Revised ©2013,
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7.1 & Graphing Exponential Transformations

3. Identifying Exponential Growth vs Decay
Exponential Equation: f(x) = Cax
C: COEFFICIENT
a: BASE
X: EXPONENT
Exponential Equations
Graph is above the x-axis
Domain: (–∞, ∞); Range: (0, ∞)
Exponential Growth
When the base is greater than 1, a > 1
f(x) is INCREASING
Exponential Decay
When the base is between zero and one, 0 < a < 1
f(x) is DECREASING
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7.1 & Graphing Exponential Transformations

4. 7.1 & 7.2 - Graphing Exponential Transformations
Example 1
Determine whether represents exponential growth or decay.
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7.1 & Graphing Exponential Transformations

5. 7.1 & 7.2 - Graphing Exponential Transformations
Example 2
Determine whether represents exponential growth or decay.
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7.1 & Graphing Exponential Transformations

6. 7.1 & 7.2 - Graphing Exponential Transformations
Your Turn
Determine whether represents exponential growth or decay.
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7.1 & Graphing Exponential Transformations

7. Graphing Exponentials
Make a table of values, usually from [–3, 3]
Plot the points from the table
Connect the dots and draw from left to right, a smooth curve and label any asymptotes
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7.1 & Graphing Exponential Transformations

8. 7.1 & 7.2 - Graphing Exponential Transformations
Example 3
Graph f(x) = 2x x
f(x) –3
(2)–3
1/8 –2
(2)–2
1/4 –1
(2)–1
1/2
(2)0 1
(2)1 2
(2)2 4 3
(2)3 8 x
f(x) –3
(2)–3 –2
(2)–2 –1
(2)–1
(2)0 1
(2)1 2
(2)2 3
(2)3 x
f(x) –3 –2 –1 1 2 3
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7.1 & Graphing Exponential Transformations

9. 7.1 & 7.2 - Graphing Exponential Transformations
Example 3
Graph f(x) = 2x x
f(x) –3
(2)–3
1/8 –2
(2)–2
1/4 –1
(2)–1
1/2
(2)0 1
(2)1 2
(2)2 4 3
(2)3 8
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7.1 & Graphing Exponential Transformations

10. 7.1 & 7.2 - Graphing Exponential Transformations
Example 3
Graph f(x) = 2x (Calculator check)
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7.1 & Graphing Exponential Transformations

11. 7.1 & 7.2 - Graphing Exponential Transformations
Example 4
Graph f(x) = (1/3)(3)x x
f(x) –3 –2 –1 1 2 3 –3
(1/3)(3)–3
1/81 –2
(1/3)(3)–2
1/27 –1
(1/3)(3)–1
1/9
(1/3)(3) 0
1/3 1
(1/3)(3) 1 2
(1/3)(3) 2 3
(1/3)(3) 3 9 –3
(1/3)(3)–2 –2 –1
(1/3)(3)–1
(1/3)(3) 0 1
(1/3)(3) 1 2
(1/3)(3) 2 3
(1/3)(3) 3
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7.1 & Graphing Exponential Transformations

12. 7.1 & 7.2 - Graphing Exponential Transformations
Your Turn
Graph f(x) = (1/2)(4)x x
f(x) –3 –2 –1 1 2 3 –3
(1/2)(4)–3
1/128 –2
(1/2)(4)–2
1/32 –1
(1/2)(4)–1
1/8
(1/2)(4) 0 1
(1/2)(4) 1 2
(1/2)(4) 2 8 3
(1/2)(4) 3 32
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7.1 & Graphing Exponential Transformations

13. 7.1 & 7.2 - Graphing Exponential Transformations
Example 5
Graph f(x) = (1/2)x x
f(x) –3
(1/2)–3 8 –2
(1/2)–2 4 –1
(1/2)–1 2
(1/2)0 1
(1/2)1
1/2
(1/2)2
1/4 3
(1/2)3
1/8 x
f(x) –3
(1/2)–3 –2
(1/2)–2 –1
(1/2)–1
(1/2)0 1
(1/2)1 2
(1/2)2 3
(1/2)3 x
f(x) –3 –2 –1 1 2 3
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7.1 & Graphing Exponential Transformations

14. 7.1 & 7.2 - Graphing Exponential Transformations
Example 5
Graph f(x) = (1/2)x x
f(x) –3
(1/2)–3 8 –2
(1/2)–2 4 –1
(1/2)–1 2
(1/2)0 1
(1/2)1
1/2
(1/2)2
1/4 3
(1/2)3
1/8
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7.1 & Graphing Exponential Transformations

15. 7.1 & 7.2 - Graphing Exponential Transformations
Example 6
Graph f(x) = –3(2/5)x (decimals are acceptable) x
f(x) –3 –2 –1 1 2 3 –3
–3(2/5)–3
–46.875 –2
–3(2/5)–2
–18.75 –1
–3(2/5)–1
–7.5
–3(2/5)0 1
–3(2/5)1
–1.2 2
–3(2/5)2
-.48 3
–3(2/5)3
-1.92
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7.1 & Graphing Exponential Transformations

16. 7.1 & 7.2 - Graphing Exponential Transformations
Your Turn
Graph f(x) = 3(2/3)x (decimals are acceptable) x
f(x) –3 –2 –1 1 2 3 –3
3(2/3)–3
10.125 –2
3(2/3)–2
6.75 –1
3(2/3)–1
4.5
3(2/3)0 3 1
3(2/3)1 2
3(2/3)2
5/3
3(2/3)3
.889
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7.1 & Graphing Exponential Transformations

17. 7.1 & 7.2 - Graphing Exponential Transformations
Translations
Vertical Translations
Moving the graph up and down; the change is OUTSIDE the exponent
Moving UP is when the translated is added, Moving DOWN is when the graph is subtracted
Horizontal Translations
Moving the graph up and down; the change is WITHIN the exponent
Moving LEFT is when the translated is added, Moving RIGHT is when the graph is subtracted
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7.1 & Graphing Exponential Transformations

18. 7.1 & 7.2 - Graphing Exponential Transformations
Example 7
If f(x)=2x, determine the transformation to g(x) = 2x State the equation for the asymptote, domain and range in Interval Notation.
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7.1 & Graphing Exponential Transformations

19. 7.1 & 7.2 - Graphing Exponential Transformations
Example 8
If f(x)=2x, determine the transformation to g(x) = 2x + 2. State the equation for the asymptote, domain and range in Interval Notation.
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7.1 & Graphing Exponential Transformations

20. 7.1 & 7.2 - Graphing Exponential Transformations
Your Turn
If f(x)=2x, determine the transformation g(x) = 2x + 3 – 4. State the equation for the asymptote, domain and range in Interval Notation.
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7.1 & Graphing Exponential Transformations

21. Vertical Stretches and Compressions
Applied to the coefficient of y = Cax
Stretches when |c| > 1
Compresses when |c| < 1
Horizontal Stretches and Compressions
Applied to the number in front of the exponent, d, of y = Cadx
Take the horizontal stretch/compression and take the reciprocal
Stretches when 0 < d < 1
Compresses when d > 1
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7.1 & Graphing Exponential Transformations

22. 7.1 & 7.2 - Graphing Exponential Transformations
Example 9
If f(x) = 2x, determine the transformation to g(x) = (1/4)(2)x
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7.1 & Graphing Exponential Transformations

23. 7.1 & 7.2 - Graphing Exponential Transformations
Example 10
If f(x) = 2x, determine the transformation to g(x) = (4)(2)x
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7.1 & Graphing Exponential Transformations

24. 7.1 & 7.2 - Graphing Exponential Transformations
Your Turn
If f(x) = 2x, determine the transformation to g(x) = (–5)(2)x
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7.1 & Graphing Exponential Transformations

25. 7.1 & 7.2 - Graphing Exponential Transformations
Example 11
If f(x) = 2x, determine the transformation to g(x) = (2)0.25x
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7.1 & Graphing Exponential Transformations

26. 7.1 & 7.2 - Graphing Exponential Transformations
Your Turn
If f(x) = 2x, determine the transformation to g(x) = (2)4x
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7.1 & Graphing Exponential Transformations

27. 7.1 & 7.2 - Graphing Exponential Transformations
Example 12
If f(x) = 2x, determine the transformation g(x) = (4) (2)x – 1 – 3. State the equation for the asymptote, domain and range in Interval Notation. Also establish if the graph is exponential growth or decay. x
f(x) –2 –1 1 2 3 x
f(x) –3
(4) (2)–3 – 1 – 3
–2.75 –2
(4) (2) –2 – 1 – 3
–2.5 –1
(4) (2)–1 – 1 – 3
(4) (2) 0 – 1 – 3 1
(4) (2)1 – 1 – 3 2
(4) (2)2 – 1 – 3 5 3
(4) (2)3 – 1 – 3 13 x
f(x) –3
(4) (2)–3 – 1 – 3 –2
(4) (2) –2 – 1 – 3 –1
(4) (2)–1 – 1 – 3
(4) (2) 0 – 1 – 3 1
(4) (2)1 – 1 – 3 2
(4) (2)2 – 1 – 3
(4) (2)x – 1 – 3 3
(4) (2)3 – 1 – 3
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7.1 & Graphing Exponential Transformations

28. 7.1 & 7.2 - Graphing Exponential Transformations
Example 12
If f(x) = 2x, determine the transformation g(x) = (4) (2)x – 1 – 3. State the equation for the asymptote, domain and range in Interval Notation. Also establish if the graph is exponential growth or decay. x
f(x) –3
(4) (2)–3 – 1 – 3
–2.75 –2
(4) (2) –2 – 1 – 3
–2.5 –1
(4) (2)–1 – 1 – 3
(4) (2) 0 – 1 – 3 1
(4) (2)1 – 1 – 3 2
(4) (2)2 – 1 – 3 5 3
(4) (2)3 – 1 – 3 13
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7.1 & Graphing Exponential Transformations

29. 7.1 & 7.2 - Graphing Exponential Transformations
Example 13
If f(x) = 2x, determine the transformation g(x) = –(1/2)x + 1 – 2. State the equation for the asymptote, domain and range in Interval Notation. Also establish if the graph is exponential growth or decay. x
f(x) –3
–(1/2)–3 + 1 – 2
–18 –2
–(1/2)–2 + 1 – 2
–10 –1
–(1/2)–1 + 1 – 2 –6
–(1/2)0 + 1 – 2 –4 1
–(1/2)1 + 1 – 2 2
–(1/2)2 + 1 – 2
–2.5 3
–(1/2)3 + 1 – 2
–2.25 x
f(x) –3 –2 –1 1 2 3
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7.1 & Graphing Exponential Transformations

30. 7.1 & 7.2 - Graphing Exponential Transformations
Your Turn
If f(x) = 2x, determine the transformation g(x) = 2(2/3)x – State the equation for the asymptote, domain and range in Interval Notation. Also establish if the graph is exponential growth or decay. x
f(x) –3
2(2/3)–3 – 1 + 3
13.125 –2
2(2/3)–2 – 1 + 3
9.75 –1
2(2/3)–
7.5
2(2/3) 6 1
2(2/3) 5 2
2(2/3)
4.33 3
2(2/3)
3.889 x
f(x) –3 –2 –1 1 2 3
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7.1 & Graphing Exponential Transformations

31. Growth/Decay Factor Equation
Equation: A = P (1 + r)t
P: Initial Principal
+: Growth Factor
-: Decay Factor
r: Interest Rate
t: Time it takes to accrue amount
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7.1 & Graphing Exponential Transformations

32. 7.1 & 7.2 - Graphing Exponential Transformations
Example 14
You invest $5,000 in an account that pays 10% interest per year. How much money will your investment be in 3 years?
A = ? Do you know much you are going to make?
P = $5,000 $5,000 is deposited
r = Interest Rate – remember it needs to be in decimal form
t = 3 Time it takes to accrue amount
+ Growth
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7.1 & Graphing Exponential Transformations

33. 7.1 & 7.2 - Graphing Exponential Transformations
Example 14
You invest $5,000 in an account that pays 10% interest per year. How much money will your investment be in 3 years?
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7.1 & Graphing Exponential Transformations

34. 7.1 & 7.2 - Graphing Exponential Transformations
Example 15
You buy a car that cost $5,000 and depreciates 10% per year. What is the value of the car after 3 years?
A = ? Do you know much you are going to make?
P = $5,000 $5,000 is spent
r = Interest Rate – remember it needs to be in decimal form
t = 3 Time it takes to accrue amount
Decay Decreasing Value
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7.1 & Graphing Exponential Transformations

35. 7.1 & 7.2 - Graphing Exponential Transformations
Example 15
You buy a car that cost $5,000 and depreciates 10% per year. What is the value of the car after 3 years?
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7.1 & Graphing Exponential Transformations

36. 7.1 & 7.2 - Graphing Exponential Transformations
Your Turn
You buy a car that cost $10,000 and depreciates 10% per year. How much money have you lost, compared to the original amount, in 5 years?
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7.1 & Graphing Exponential Transformations

37. 7.1 & 7.2 - Graphing Exponential Transformations
Assignment
Pg 482: 3-5 all, Graph, Label domain and range, and asymptotes for 7-23 EOO, 24, 25, 27, 28, 30 Pg 489: 3-6 all, Graph, Label domain and range, and asymptotes for 7-23 EOO, 26, 27, 31a, 31c
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7.1 & Graphing Exponential Transformations