Exponential Functions Section 3.1 Precalculus PreAP/Dual, Revised ©2018 viet.dang@humbleisd.net 12/30/2018 3:35 AM §3.1: Exponential Functions and Graphs

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

§3.1: 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

§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

§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

§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

§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

§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

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

§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

§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

§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

§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

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

§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

§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

§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

§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

§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

§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 ½ 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

§3.1: Exponential Functions and Graphs Assignment Page 208 13-16 all, 17-21 odd, 27-30 all 12/30/2018 3:35 AM §3.1: Exponential Functions and Graphs