1. Exponential Growth & Decay Graphs
I.. Exponential Functions.
A) Exponential function: y = bx (parent function)
1) If b > 1, then it is an exponential Growth function.
2) If 0 < b < 1, then its an expo Decay function
a) Or if b > 1 but y = b-x (the neg expo flips the b term).
3) The “b” term is called the growth factor.
Expo Expo.
growth decay
graph graph

2. Exponential Growth & Decay Graphs
II.. Exponential Growth Function.
A) Parent function (for growth): y = bx (where b > 1)
B) Table of value for y = 2x (an example growth function)
C) Growth graphs rises to the right, flatten to the left.
as x  –∞, f(x)  as x  ∞, f(x) = +∞
D) Has a horizontal asymptote at y = 0.
E) If x = 0, then bx = b0 which = 1.
1) Passes through (0, 1) (0, 1)
Let’s call this the Critical Point.
This is what moves in shift rules. x –4 –3 –2 –1 1 2 3 4 y
1/16
1/8
1/4
1/2 8 16

3. Exponential Growth & Decay Graphs
I.. Exponential Decay Function.
A) Parent decay function: y = bx
(where 0 < b < 1)
1) Or if b >1 but y = b-x
B) The “b” term is called the Decay Factor.
C) Decay graph: rises to the left, flattens to the right.
as x  –∞, f(x)  +∞ as x  +∞, f(x) = 0
D) Has a horizontal asymptote at y = 0.
E) If x = 0, then b0 = 1
1) Passes thru ( 0 , 1 ) = the critical point.
(0 , 1)

4. Exponential Growth & Decay Graphs
III.. Shifts of Exponential Graphs.
A) y = bx + d
1) Moves the graph up/down. + is up, – is down.
2) It also moves the horizontal asymptote up/down: y = d.
B) y = bx+c (the exponent is a grouping symbol)
1) Moves the graph left/right. + goes , – goes
2) Set the exponent = 0, solve for x for sideways shift.
C) y = a • bx (the “a” term is the “slope”)
1) If a > 1, then the graph gets “taller” [vert. stretch].
2) If 0 < a < 1, then the graph gets “less tall” [vert. shrink].
3) If a is negative, the graph flips (over the x-axis).

5. Exponential Growth & Decay Graphs
IV.. Sketching Shifted Exponential Functions.
y = a • bx+c + d
A) State all the shifts.
B) Find the new critical point.
1) New critical pt = (sideways shift , add first + last terms) or (–c , a+d)
2) If y = bx+c + d, then the “a” term is 1. (understood 1• bx+c + d)
C) Move the horizontal asymptote (y = d) and draw it.
D) Look at the “slope” (the “a” term).
1) Stretched graphs have a crit pt far from the horiz asy.
2) Shrunk graphs have a crit pt close to horiz asy.
3) Flip graphs are below the horizontal asymptote line (they flip).

6. Exponential Growth & Decay Graphs
Examples: Sketch these functions.
y = (5)4x– ) y = (1/2)3x+1 – 2
vert stretch vert shrink
crit pt = (3, 5 + 2) = (3,7) crit pt = (–1, ½ – 2) = (–1, –1.5)
horiz asy: y = horiz asy: y = – 2
(3,7)
(–1, –1.5)

7. Exponential Growth & Decay Graphs
Examples: Sketch these functions.
3) y = (–3)5x+2 – ) y = (– 4/5)4x–2 + 3
flip taller flip shorter
crit pt = (–2, –3 – 1) = (–2, –4) crit pt = (2, – 4/5 + 3) = (2, 2.2)
horiz asy: y = – horiz asy: y = 3
(2, 2.2)
(–2, –4)