1. 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

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

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

4. 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, =1

5. 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.

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

7. 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.

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

9. 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!

10. Basic Exponential Graphs Worksheet

11. 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

12. 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.

13. 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!

14. 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!

15. 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!

16. 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

17. Examples
𝑦= 2 𝑥 𝑦=− 2 𝑥

18. Examples
𝑦= 2 𝑥 𝑦= 2 −𝑥

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

20. 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