1. Exponential Functions
f(x) = 4(2)x
What is the initial value?
What is the growth factor?
What is the domain?
By definition if a is the growth factor then any term divided by the previous term should get you a.

2. Exponential Graph parent function ax where a > 1
Properties:
The domain is (-∞,∞)
The range is (0,∞)
There are no x intercepts (asymptote) as x  -∞ f(x) 0
The y intercept is (0,1) because anything to the 0 power is 1
The points (-1,1/a), (0,1), (1,a) are on the graph.
Graph is smooth and continuous.

3. Exponential Graph parent function ax where 0 < a < 1
Properties:
The domain is (-∞,∞)
The range is (0,∞)
There are no x intercepts (asymptote) as x  ∞ f(x) 0
The y intercept is (0,1) because anything to the 0 power is 1
The points (-1,1/a), (0,1), (1,a) are on the graph.
Graph is smooth and continuous.

4. y = Cbx - h + k
To be an exponential growth b must be greater than 1.
The C, h, and k do the same things as they always have. a stretches and flips, h moves it left and right and k moves it up and down. This effects the asymptote and the critical point how?
Find 2 points usually whatever makes the exponent = 0 and 1. and the asymptote moves with k.

5. y = 3x+1 – 2

6. y = -2(4)x-3

7. The horizontal asymptote shifts down 2 as does the critical point.
y = 1/3x+1 – 2
Let x = -1 since = 0
And solve:
When x = -1 (1/3) = 1 – 2 = -1
Let x = 0
And solve:
When x = 0 (1/3) = 1/ = -1 2/3

8. The horizontal asymptote shifts up1 as does the critical point
The horizontal asymptote shifts up1 as does the critical point. It also flips
y = -1/3x +1
Let x = 0
And solve:
When x = 0 (-1/3)0 = 1
Let x = 1
And solve:
When x = 1 (-1/3)1 = -1/3

9. Practice: y = -3x +1

10. Natural base

11. y = 2ex+1 – 2

12. y = e-2x + 1

13. Solving Exponential equations
If you have an equation and the bases are the same then what must be true about the exponents for the equation to remain equal?
The exponents must also be equal.
So if 2x+1 = 23x-5 then x + 1 = 3x – 5 and we can solve for x.
Brilliant deduction my dear Watson.

14. But what if the bases are not the same?
We make them the same using exponents. Changing the larger base to a smaller base raised to an exponent
Elementary

15. Your turn to try!! Remember your exponent rules.
Hint: How can you change 1/9 to 9 using exponents?