1. Graphing y = ebx Functions
I.. The Natural Base e.
A) e ≈ 2.72
B) “e” occurs in nature and in math/science formulas.
C) e = as “n” approaches + ∞.
D) y = A•ebx+C + D is the natural base exponential function.
E) y = ebx is the parent function: critical pt at (0 , 1)
1) + b = Growth graph ) – b = Decay graph
3) Horizontal asymptote: y = 0.

2. Graphing y = ebx Functions
II.. Natural Base Exponential Function Shifts y = A•ebx+C + D
A) y = ebx + D “D” moves the graph up (+) or down (–).
1) It also moves the horizontal asymptote: y = D.
B) y = ebx+C moves the graph sideways.
1) Set the exponent part = 0 and solve for x (Bx + C = 0)
That is the sideways shift.
C) y = A•ebx the “A” term is the “slope”.
1) A > 1 is V stretch 2) 0< A <1 is V shrink 3) –A flips over x-axis
D) y = ebx the sign of the “b” term determines growth/decay
1) +b = growth graph 2) –b = decay graph

3. Graphing y = ebx Functions
III.. Sketching Natural Base e Functions ( y = A•ebx+C + D ).
A) State all the shifts.
B) Find the new critical point (0 , 1)  ( # , # )
1) new crit pt = (sideways shift, first # + last #)
or ( bx + C = 0 , A + D )
C) The new horizontal asymptote is y = D.
D) Look at the “slope” (A term) to see if it is a flip graph or not.
E) Determine if it is a Growth or Decay graph (the “b” sign).
F) Sketch the horizontal asy. and the critical point.
1) Draw the Growth / Decay (or flipped) graph from crit pt.

4. Graphing y = ebx Functions
Examples:
1) y = 3e4x+8 – ) y = ½e–4x
4x + 8 = Growth – 4x + 12 = Decay
4x = – – 4x = –12
x = – x = 3
steeper less steep
crit pt = ( – 2 , – 3 ) crit pt = ( 3 , 4.5)
H asy: y = – H asy: y = 4
(3 , 4.5)
(–2 , –3)

5. Graphing y = ebx Functions
IV.. Properties of Exponents
A) xm • xn = xm+n (Add the exponents)
B) (Subtract the exponents)
C) (Multiply the exponents)
D) and (Negative expo flip)
E) (Simplify radicals)

6. Graphing y = ebx Functions
V.. Evaluating ex with a calculator.
A) There are two “e” buttons on the TI-84 calculators.
* 1) 2nd ex displays e^( on the screen.
2) 2nd e displays e on the screen.
B) The Nspire has the button ex.
Example: Find 3e4 + 7
On the TI On the Nspire
3 • e^(4) + 7 = e4 + 7 =