0401 ch 4 day 1

Chapter 4

Exponential & Logarithmic
Functions

Proverbs 24:14 Know that wisdom is such to your
soul; if you ﬁnd it, there will be a future, and your
hope will not be cut off.

4.1 Exponential Functions

A function that can be expressed in
the form y = b , b > 0 and b ≠ 1 is
x

called an exponential function.


A function that can be expressed in
the form y = b , b > 0 and b ≠ 1 is
x

called an exponential function.

These functions are used to model
situations such as growth and decay.


x
y=b


x
y=b

the base of the
exponent


x
y=b
our input
variable
the base of the
exponent


Graph and discuss
x x
y1 = 2 y2 = 5


Graph and discuss
x x
y1 = 2 y2 = 5

Domain:


Graph and discuss
x x
y1 = 2 y2 = 5

Domain: {x : x ∈° }


Graph and discuss
x x
y1 = 2 y2 = 5

Range:


Graph and discuss
x x
y1 = 2 y2 = 5

Range: {y : y > 0}


Graph and discuss
x x
y1 = 2 y2 = 5

Range: {y : y > 0}
x-intercept:


Graph and discuss
x x
y1 = 2 y2 = 5

Range: {y : y > 0}
x-intercept: none y = 0 is a H.A.


Graph and discuss
x x
y1 = 2 y2 = 5

Range: {y : y > 0}
y-intercept:


Graph and discuss
x x
y1 = 2 y2 = 5

Range: {y : y > 0}
y-intercept: ( 0,1)


Graph and discuss
x x
y1 = 2 y2 = 5

Range: {y : y > 0}
y-intercept: ( 0,1)
increasing


Graph and discuss
x x
y1 = 2 y2 = 5

Range: {y : y > 0}
y-intercept: ( 0,1)
increasing
1 to 1


Let’s experiment with various b values by graphing
x x x
y=8 y = .7 y =1


x x x
y=8 y = .7 y =1
as b → ∞ ... grows faster


x x x
y=8 y = .7 y =1
b >1 ... increasing


x x x
y=8 y = .7 y =1
b >1 ... increasing
0 < b <1 ... decreasing


x x x
y=8 y = .7 y =1
b >1 ... increasing
0 < b <1 ... decreasing
b =1 ... constant (not exponential)


x
⎛ 1 ⎞
Graph y1 = 2 and
x
y2 = ⎜ ⎟
⎝ 2 ⎠


x
⎛ 1 ⎞
Graph y1 = 2 and
x
y2 = ⎜ ⎟
⎝ 2 ⎠

they are symmetric about the y-axis


x
⎛ 1 ⎞
Graph y1 = 2 and
x
y2 = ⎜ ⎟
⎝ 2 ⎠

both have a y-intercept of ( 0,1)


x
⎛ 1 ⎞
Graph y1 = 2 and
x
y2 = ⎜ ⎟
⎝ 2 ⎠

both have a y-intercept of ( 0,1)
x
−x 1 ⎛ 1 ⎞
note : 2 = x = ⎜ ⎟
2 ⎝ 2 ⎠


x
Graph y1 = e (the Natural Exponential Function)


x

e is a constant ... irrational ... like π or 2


x

n
⎛ 1 ⎞
e = ⎜ 1+ ⎟ as n → ∞
⎝ n ⎠


x

n
⎛ 1 ⎞
e = ⎜ 1+ ⎟ as n → ∞
⎝ n ⎠
1
do e to see that e ≈ 2.7183


x

n
⎛ 1 ⎞
e = ⎜ 1+ ⎟ as n → ∞
⎝ n ⎠
1
do e to see that e ≈ 2.7183
e is the base of natural logarithms
(more on this later)


Predict, verify with a graph, then discuss


x
1) y = 2 + 3


x
1) y = 2 + 3
x
2) y = −e


x
1) y = 2 + 3
x
2) y = −e
x
⎛ 1 ⎞
3) y = ⎜ ⎟ − 4
⎝ 3 ⎠


x
1) y = 2 + 3
x
2) y = −e
x
⎛ 1 ⎞
3) y = ⎜ ⎟ − 4
⎝ 3 ⎠

Be sure to read Example 8 carefully ...
do it on your calculator!


HW #1
Striving for success without hard work is like
trying to harvest where you haven’t planted.
David Bly

0401 ch 4 day 1

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