Warm Up – 4.4 - Friday State the transformations that have occurred on each function. 𝑦=3 sin 2𝑥 −3 𝑦=− cos 5(𝑥−𝜋 )+2

Negative Exponents 3 −1 = 1 3 3 −2 = 1 3 2 3 −3 = 1 3 3 3 −1 = 1 3 3 −2 = 1 3 2 3 −3 = 1 3 3 1 2 −1 = 2 1 1 1 2 −3 = 2 1 3 2 3 −1 = 3 2 1 2 3 −3 = 3 2 3

Negative Exponents 1 2 −1 = 2 1 1 1 2 −3 = 2 1 3 1 2 −1 = 2 1 1 1 2 −3 = 2 1 3 2 3 −1 = 3 2 1 2 3 −3 = 3 2 3

Zero Rule Anything to the Zero power is 1. 2 0 =1 −5 0 =1 2 0 =1 −5 0 =1 8,234,517,689,065,230,124,521,009 0 =1

Exponential Functions An exponential function is any function of the form: 𝑦=𝑎∙ 𝑏 𝑥 . An exponential function is called exponential because X is in the exponent. This is different from a power function or polynomial where x is raised to an exponent.

Growth vs. Decay 𝐺𝑟𝑜𝑤𝑡ℎ:𝑎>0 𝐷𝑒𝑐𝑎𝑦:0<𝑎<1

Asymptotes A Horizontal Asymptote is an imaginary line that an exponential function approaches but never reaches as the y-value gets smaller and smaller. This graph has a horizontal Asymptote at 𝑦=0.

Plotting Basic Exp. Graphs Graph the following functions. Then state the domain, range, and y-intercept. 𝑦=2 4 𝑥 𝑦=3 1 2 𝑥

Rules for Exponentials The domain for an exponential should always be All Real Numbers. The range is determined by where the Asymptote is. We get the y-intercept by plugging in 0!

Basic Exponential Graphs Worksheet

Transforming Exponential Graphs The Rules are the same as always! 𝐵𝑎𝑠𝑖𝑐 𝐺𝑟𝑎𝑝ℎ: 𝑦=𝑎 𝑏 𝑥 Vertical Shift: 𝑦=𝑎 𝑏 𝑥 +𝑐 Positive Up, Negative Down 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑆ℎ𝑖𝑓𝑡:𝑦=𝑎 𝑏 𝑥+𝑐 Positive Left, Negative Right

Horizontal Asymptotes Horizontal Asymptotes work the exact same way zero lines did for our trig functions. Normally it starts at zero, but if I add a number it moves up and if I subtract it moves down.

Examples 𝑦= 2 𝑥 𝑦= 2 𝑥 +5 Range: 𝑦>0 Range: 𝑦>4 𝑦= 2 𝑥 𝑦= 2 𝑥 +5 Range: 𝑦>0 Range: 𝑦>4 Notice our y-intercept also went up 5!

Examples 𝑦= 2 𝑥 𝑦= 2 (𝑥+2) Range: 𝑦>0 Range: 𝑦>0 𝑦= 2 𝑥 𝑦= 2 (𝑥+2) Range: 𝑦>0 Range: 𝑦>0 To get the new y-intercept, plug in 0 for X!

Transforming Exponential Graphs The Rules are the same as always! 𝐵𝑎𝑠𝑖𝑐 𝐺𝑟𝑎𝑝ℎ: 𝑦=𝑎 𝑏 𝑥 Vertical Stretch: 𝑦=𝐶(𝑎 𝑏 𝑥 ) 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑆𝑡𝑟𝑒𝑡𝑐ℎ:𝑦= 𝑎𝑏 𝐶𝑥 Remember Horizontal Stretches are backwards. Multiplying by 2 is a compression while multiplying by ½ is a stretch!

Transforming Exponential Graphs The Rules are the same as always! 𝐵𝑎𝑠𝑖𝑐 𝐺𝑟𝑎𝑝ℎ: 𝑦=𝑎 𝑏 𝑥 Vertical Reflection: 𝑦=−𝑎 𝑏 𝑥 Positive Up, Negative Down 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛:𝑦= 𝑎𝑏 −𝑥 Positive Left, Negative Right

Examples 𝑦= 2 𝑥 𝑦=− 2 𝑥

Examples 𝑦= 2 𝑥 𝑦= 2 −𝑥

Example State what transformations have been applied to 𝑦= 2 𝑥 . A) 𝑦=3 2 𝑥 −2 B) 𝑦=− 2 2(𝑥−1)

Example Solutions 𝐴) 𝑦=3 2 𝑥 −2 Vertical Stretch by 3 Vertical Shift Down 2 B) 𝑦=− 2 2(𝑥−1) Horizontal Compression by 2 Horizontal Shift Right 1 Vertical Reflection