UNIT #1: Transformation of Functions; Exponential and Log

Goals:

• Review core function families and mathematical transformations.

Textbook reading for Unit #1: Read Sections 1.1–1.4

2

Example: The graphs of ex, ln(x), x2 and x12 are shown below. Identify

each function’s graph.

x

y1

Unit 1 – Transformation of Functions; Exponential and Log 3

Comment on the properties of the graphs of

• inverse functions -

• exponentials -

• logarithms -

• powers of x -

4

Knowing the graphs and properties of essential families of functions is crucial foreffective mathematical modeling.Name other families of functions.

Unit 1 – Transformation of Functions; Exponential and Log 5

Give examples of members of each family, and state some of their common

properties.

6

The core families of functions can be made even more versatile by being trans-

formed.Example: Sketch the graph of y = x2, over the interval x ∈ [−4, 4].

On the same axes, sketch the graph of y = 4− 12(x + 1)2.

Unit 1 – Transformation of Functions; Exponential and Log 7

Review the four common types of function transformations.Type Form Example

8

Type Form Example

Unit 1 – Transformation of Functions; Exponential and Log 9

Example: Consider the data shown below, showing the concentration of a

chemical produced in a reaction vessel, over time.

0 20 40 60 80 100

05

1015

Time (hours)

Con

cent

ratio

n (p

pm)

What family of functions would best describe this graph? Point out specific

features of the graph that make the choice a reasonable one.

10

Give a general mathematical form for the func-

tion, based on the shape of the graph.

e.g. C(t) = ...

0 20 40 60 80 100

05

1015

Time (hours)

Con

cent

ratio

n (p

pm)

Unit 1 – Transformation of Functions; Exponential and Log 11

Determine as many of the numerical values in the

formula C(t) = ... as you can, given the graph.

Sketching related graphs along the way might be

helpful.

0 20 40 60 80 100

05

1015

Time (hours)

Con

cent

ratio

n (p

pm)

12

Looking closely at the graph, you see that after

30 hours, the concentration has reached almost

exactly 12 ppm. Determine the value for the fi-

nal missing parameter in your concentration func-

tion.0 20 40 60 80 100

05

1015

Time (hours)

Con

cent

ratio

n (p

pm)

Unit 1 – Transformation of Functions; Exponential and Log 13

Logarithm Review

Most students are quite comfortable with exponential functions, but many findlogarithms less familiar. To address this we will do a more comprehensive reviewof the logarithmic function and its use in transforming equations.

Log/Exponential Equivalency

ac = x means loga x = c

Simplify loga(a7).

Simplify aloga(25).

14

Without using a calculator, find log10(1/100) and log10(10, 000) .

Unit 1 – Transformation of Functions; Exponential and Log 15

These problems suggest the following equations, which also follow from the factthat ax and loga(x) are inverse functions.

loga(ax) = x and aloga x = x

Rules for Computing with Logarithms

1. loga(AB) = logaA + logaB

2. loga(A/B) = logaA− logaB

3. loga(AP ) = P logaA

16

Changing logarithmic bases

The functions ax and loga are not provided on calculators unless a = 10 or a = e(see next section of these notes). For other values of a, ax and loga can be expressedin terms of 10x and log10. To calculate loga x, we use the following formula:

Conversion of Log Bases

loga x =log x

log a

Prove the above formula, using the Rules for Computing Logarithms and the

fact that loga x = c means x = ac.

Unit 1 – Transformation of Functions; Exponential and Log 17

Graphs of Logarithmic FunctionsSince the logarithm in base 10 is commonly used in science, we define log x (nosubscript) to mean log10 x, for brevity.

The graph of log x may be obtained from the graph of 10x by reversing the axes(that is, by reflecting the graph in the line y = x). (If drawing the graph of inversefunctions is unfamiliar, please read Section 1.3 in the text.)

18

10

10

10x

log10(x)

What is the domain of log x? What is the range of log x?

Sketch the logarithm function for the bases e and 2.

Unit 1 – Transformation of Functions; Exponential and Log 19

Classic Applications of Exponentials and Logarithms

Example: Based on H-H, Section 1.4 #48: A cup of coffee contains 100

mg of caffeine, which leaves the body at a continuous rate of 17% per hour.

Sketch the graph of caffeine level over time, after drinking one cup of coffee.

20

There are two natural interpretations of the question statement which lead to

two different formulae for A(t). Write down both formulae.

Compare the predicted caffeine level after 10 hours, using each model. Based

on those values, how similar are these two models in practice?

Unit 1 – Transformation of Functions; Exponential and Log 21

The key phrase continuous rate has a special meaning in mathematics and science,and it associated with the natural exponential form ert. It is typically associatedwith processes like chemical reactions, population growth, and continuously com-pounded interest.Common alternative statements about percentage growth or decay, where the rateis assumed to be measured at the end of one time period (hour, day year), areusually of the form (1± r)t.

22

Write out an appropriate mathematical model for the following scenarios:

• Infant mortality is being reduced at a rate of 10% per year.

• My $10,000 investment is growing at 5% per year.

• A savings account offers daily compound interest, at a 4% annual rate.

• Bacteria are reproducing at a continuous rate of 125% per hour.

Unit 1 – Transformation of Functions; Exponential and Log 23

We now return to our earlier modeling problem.Example: A cup of coffee contains 100 mg of caffeine, which leaves the

body at a continuous rate of 17% per hour. Write the formula for A(t).

What is the caffeine level at t = 4 hours?

At what time does the caffeine level reach A = 10 mg?

Find the half-life of caffeine in the body.