1. Exponential Functions
Section 3.1
Precalculus PreAP/Dual, Revised ©2018
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

2. Exponential Growth vs Decay
Exponential Equation: 𝒇 𝒙 =𝑪 𝒂 𝒙
𝑪: COEFFICIENT
𝒂: BASE
𝑿: EXPONENT
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
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

3. §3.1: Exponential Functions and Graphs
Example 1
Determine whether 𝒇 𝒙 = 𝟏 𝟓 𝒙 represents exponential growth or decay.
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

4. §3.1: Exponential Functions and Graphs
Your Turn
Determine whether 𝒇 𝒙 = 𝟏𝟎 𝟒 𝟑 𝒙 represents exponential growth or decay.
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

5. §3.1: Exponential Functions and Graphs
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
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

6. §3.1: Exponential Functions and Graphs
Example 2
If 𝒇 𝒙 = 𝟐 𝒙 , determine the transformation to 𝒈 𝒙 = 𝟐 𝒙 +𝟐
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

7. §3.1: Exponential Functions and Graphs
Example 3
If 𝒇 𝒙 = 𝟐 𝒙 , determine the transformation to 𝒈 𝒙 = 𝟐 𝒙+𝟐
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

8. §3.1: Exponential Functions and Graphs
Your Turn
If 𝒇 𝒙 = 𝟐 𝒙 , determine the transformation to 𝒈 𝒙 = 𝟐 𝒙+𝟑 −𝟒
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

9. Vertical Stretches and Compressions
Applied to the coefficient of 𝒚=𝑪 𝒂 𝒙
Stretches when 𝑪 >𝟏
Compresses when 𝑪 <𝟏
Horizontal Stretches and Compressions
Applied to the number in front of the exponent, 𝒅, of 𝒚=𝑪 𝒂 𝑫𝒙
Take the horizontal stretch/compression and take the reciprocal
Stretches when 𝟎<𝒅<𝟏
Compresses when 𝒅>𝟏
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

10. §3.1: Exponential Functions and Graphs
Example 4
If 𝒇 𝒙 = 𝟐 𝒙 , determine the transformation to 𝒈 𝒙 = 𝟏 𝟒 𝟐 𝒙
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

11. §3.1: Exponential Functions and Graphs
Your Turn
If 𝒇 𝒙 = 𝟐 𝒙 , determine the transformation to 𝒈 𝒙 =(−𝟓) 𝟐 𝒙
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

12. §3.1: Exponential Functions and Graphs
Example 5
If 𝒇 𝒙 = 𝟐 𝒙 , determine the transformation to 𝒈 𝒙 = 𝟐 𝟎.𝟐𝟓𝒙
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

13. §3.1: Exponential Functions and Graphs
Your Turn
If 𝒇 𝒙 = 𝟐 𝒙 , determine the transformation to 𝒈 𝒙 = 𝟐 𝟒𝒙
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

14. 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
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

15. §3.1: Exponential Functions and Graphs
Example 6
Graph 𝒇 𝒙 = 𝟐 𝒙 𝒙
𝒇(𝒙) −𝟑
𝟐 −𝟑
𝟏/𝟖 −𝟐
𝟐 −𝟐
𝟏/𝟒 −𝟏
𝟐 −𝟏
𝟏/𝟐 𝟎
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 𝟎
𝟐 𝟎 𝟏
𝟐 𝟏 𝟐
𝟐 𝟐 𝟑
𝟐 𝟑 𝒙
𝒇 𝒙 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

16. §3.1: Exponential Functions and Graphs
Example 6
Graph 𝒇 𝒙 = 𝟐 𝒙 𝒙
𝒇(𝒙) −𝟑
𝟐 −𝟑
𝟏/𝟖 −𝟐
𝟐 −𝟐
𝟏/𝟒 −𝟏
𝟐 −𝟏
𝟏/𝟐 𝟎
𝟐 𝟎 𝟏
𝟐 𝟏 𝟐
𝟐 𝟐 𝟒 𝟑
𝟐 𝟑 𝟖
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

17. §3.1: Exponential Functions and Graphs
Example 6
Graph 𝒇 𝒙 = 𝟐 𝒙 (Calculator check)
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

18. §3.1: Exponential Functions and Graphs
Example 7
Graph 𝒇 𝒙 = 𝟏 𝟐 𝒙 𝒙
𝒇 𝒙 −𝟑
𝟏/𝟐 −𝟑 𝟖 −𝟐
𝟏/𝟐 −𝟐 𝟒 −𝟏
𝟏/𝟐 −𝟏 𝟐 𝟎
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 𝟎
𝟏/𝟐 𝟎 𝟏
𝟏/𝟐 𝟏 𝟐
𝟏/𝟐 𝟐 𝟑
𝟏/𝟐 𝟑 𝒙
𝒇 𝒙 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

19. §3.1: Exponential Functions and Graphs
Example 7
Graph 𝒇 𝒙 = 𝟏 𝟐 𝒙 𝒙
𝒇 𝒙 −𝟑
𝟏/𝟐 −𝟑 𝟖 −𝟐
𝟏/𝟐 −𝟐 𝟒 −𝟏
𝟏/𝟐 −𝟏 𝟐 𝟎
𝟏/𝟐 𝟎 𝟏
𝟏/𝟐 𝟏
𝟏/𝟐
𝟏/𝟐 𝟐
𝟏/𝟒 𝟑
𝟏/𝟐 𝟑
𝟏/𝟖
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

20. §3.1: Exponential Functions and Graphs
Your Turn
Graph 𝒇 𝒙 = 𝟏 𝟐 𝟒 𝒙 𝒙
𝒇(𝒙) −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑 −𝟑
(𝟏/𝟐) (𝟒) −𝟑
𝟏/𝟏𝟐𝟖 −𝟐
(𝟏/𝟐) (𝟒) −𝟐
𝟏/𝟑𝟐 −𝟏
(𝟏/𝟐) (𝟒) −𝟏
𝟏/𝟖 𝟎
(𝟏/𝟐) (𝟒) 𝟎 𝟏
(𝟏/𝟐) (𝟒) 𝟏 𝟐
(𝟏/𝟐) (𝟒) 𝟐 𝟖 𝟑
(𝟏/𝟐) (𝟒) 𝟑 𝟑𝟐
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs

21. §3.1: Exponential Functions and Graphs
Assignment
Page all, odd, all
12/30/2018 3:35 AM
§3.1: Exponential Functions and Graphs