Unit #1 : Transformation of Functions, Exponentials and Loga-rithms

Goals:

• Review core function families and mathematical transformations.

Familiar Functions - 1

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

each function’s graph.

x

y (a) (b)

(c)

(d)

(a)

(b)

(c)

(d)

Familiar Functions - 2

Comment on the properties of the graphs of

• inverse functions -

• exponentials -

• logarithms -

• powers of x -

Familiar Functions - 3

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

Familiar Functions - 4

Give examples of members of each family.

Transforming Functions - 1

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.

Transforming Functions - 2

Review the four common types of function transformations.Type Form ExampleVertical Shift

Vertical Scaling

Transforming Functions - 3

Type Form ExampleHorizontal Shift

Horizontal Scaling

Modeling With Transformations - 1

Example: Consider the data shown below, showing the concentration of achemical produced in a reaction vessel, over time.

0

5

10

15

0 20 40 60 80 100t

C(t)

b

b

bb

bb

bb b b b b b b b b b b b b b

Time (hours)

Con

centration(ppm)

What family of functions would best describe this graph? Point out specificfeatures of the graph that make the choice a reasonable one.

Modeling With Transformations - 2

Give a general mathematical form for the func-tion, based on the shape of the graph.e.g. C(t) = ...

0

5

10

15

0 20 40 60 80 100t

C(t)

b

b

bb

bb

bb b b b b b b b b b b b b b

Time (hours)

Con

centration(ppm)

Modeling With Transformations - 3

Determine as many of the numerical values in theformula C(t) = ... as you can, given the graph.Sketching related graphs along the way might behelpful. 0

5

10

15

0 20 40 60 80 100t

C(t)

b

b

bb

bb

bb b b b b b b b b b b b b b

Time (hours)

Con

centration(ppm)

Modeling With Transformations - 4

Looking closely at the graph, you see that after30 hours, the concentration has reached almostexactly 12 ppm. Determine the value for the fi-nal missing parameter in your concentration func-tion. 30

12

0

5

10

15

0 20 40 60 80 100t

C(t)

b

b

bb

bb

bb b b b b b b b b b b b b b

Time (hours)

Con

centration(ppm)

Logarithm Definition - 1

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).

Logarithm Definition - 2

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

Logarithm Definition - 3

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

Changing Log Bases - 1

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 =log10 x

log10 aor

loge x

loge a

Prove the above formula, using the Rules for Computing Logarithms and thefact that loga x = c means x = ac.

Changing Log Bases - 2

Example: Without an exact calculation, determine which of log10 1000 andlog2 1000 is the larger numeric value.

Compute the numeric value of both the log values above, using your calculatorif necessary.

Note: since the logarithm in base 10 is commonly used in science, we define log x(no subscript) to mean log10 x, for brevity.For the natural logarithm (base e), we use “ln” instead of “loge”.

Graphs of Logarithmic Functions - 1

Graphs of Logarithmic Functions

The graph of loga x may be obtained from the graph of its corresponding expo-nential function by reversing the axes (that is, by reflecting the graph in the liney = x).E.g. for y = log10 x and y = 10x,

x

y

2

4

6

8

10

−2

−4

2 4 6 8 10−2−4

x

y

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

Graphs of Logarithmic Functions - 2

Example: Sketch the logarithm function for the bases e and 2.

x

y

2

4

6

−2

−4

2 4 6−2−4

x

y

Applications of Exponentials and Logarithms - 1

Applications of Exponentials and Logarithms

Example: A cup of coffee contains 100 mg of caffeine, which leaves thebody at a continuous rate of 17% per hour.

Sketch the graph of caffeine amount over time, A(t), after drinking one cupof coffee.

t

A(t)

Applications of Exponentials and Logarithms - 2

There are two natural interpretations of the question statement which lead totwo different formulae for A(t). Write down both formulae.

Compare the predicted caffeine level after 10 hours, using each model. Basedon those values, how similar are these two models in practice?

Translating Rates Into Formulas - 1

Translating Rates Into Formulas

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.

Translating Rates Into Formulas - 2

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.

Translating Rates Into Formulas - 3

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

Translating Rates Into Formulas - 4

Find the half-life of caffeine in the body.

Sketch a more detailed graph of the level of caffeine in the body over time.

t

A(t)