PreCalculus: Lesson Plan * 10/27/2006

Quickstart
Block I
p.245 #96
Human Memory Model

Lesson
3.4 Solving Exponential and Logarithmic Equations
Objectives: Students will know how to solve exponential and logarithmic equations.

I. Introduction and Review (p.247)
A. One-to-One Properties
a^x = a^y if and only if x = y
log_ax = log_ay if and only if x = y.
B. Inverse Properties
log_aa^x = x
a^log_ax = x

II. Solving Exponential Equations (pp. 249 - 249)
Two very general keys to solving exponential equations are:
1. Isolate the exponential expression
2. Use the second one-to-one property from above.
See Examples 3, 4, 5

IV. Application
See Example 12 in textbook about Doubling an Investment
Example 5: Saving for Retirement
You have $50,000 to invest. You need to have $350,000 to retire in thirty years. At what continuously compounded interest rate would you need to invest to reach your goal?
A = Pe^rt
350,000 = 50,000 e^30r
7 = e^30r
ln 7 = ln e^30r
ln 7 = 30r
r = ln 7 / 30 ≈ 6.5%

Check out The Mint to find out about money management for kids.

Also check out the website sponsored by the Federal Deposit Insurance Corporation (FDIC)
Start Smart - Money Management for Teens

Lesson Plan: PreCalculus * 10/26/2006

High School Professional Day

Assignment
Worksheet
Review p.243, 244, 245

Each page is one grade.
Due Monday, October 30.

PreCalculus: Lesson Plan * 10/25/2006

Quickstart
Writing About Math p.235
Human Memory Model

Lesson
3.3 Properties of Logarithms (continued)
Objective: Students will know how to rewrite logarithmic functions with a different base, use properties of logarithms to evaluate, rewrite, expand, and condense logarithmic expressions.

III. Rewriting Logarithms (continued)
Example 3
Condense the logarithmic expression
a) 2log - 3logy + 1/2·logz
= log x² - log y³ + log√z
= log (x²/y³) + log√z
= log (x²√z/y³)
b) 1/3(2lnx - 4 ln y - ln(z + 2)
= 1/3(lnx² - ln y⁴ - ln(z+2))
= 1/3[ln(x²/[y⁴(z + 2)])]
= ln ³√x²/[y⁴(z + 2)]

IV Application
Example 4: On the Richter Scale, the magnitude R of an earthquake of intensity I is given by
R = (lnI - lnI₀)/ln 10, where I₀ is the minimum intensity used for comparison. Write this as a single common logarithmic expression.
R = (lnI - lnI₀)/ln 10
= (log I/log e) - (log I₀ / log e)
——————————————————— Using Change of Base formula
(log 10/log e)
= log I - log I₀
——————— Multiply by log e / log e
log 10
= log I - log I₀ log 10 = 1
= log (I / I₀) Quotient property

Read about and watch a video news report about the magnitude 6.0 earthquake that struck in the Gulf of Mexico September 10th, 2006.

This map shows where people have reported to the United States Geological Survey (USGS) that they felt the earthquake.

Homework
p.244 - 246 #3 - 67 odd

Algebra I: Lesson Plan * 10/24/2006

Chapter 5: Graphs and Functions 5-2: Relations and Functions Objective: To identify and describe relations and functions. To evaluate functions using variables.

Quickstart
MA.B.2.4.2 Practice Worksheet

Lesson
Identifying Relations and Functions p.241 - 242
Example 1: Finding Domain and Range
Definition of a Function
Example 2: Using the Vertical-Line Test

This picture, from freeimages.co.uk, shows a pencil. You can use a pencil as a model of a vertical-line for the vertical-line test.

Example 3: Using a Mapping Diagram
Check Understanding #1 - 3

Evaluating Functions p.243
Example 4: Evaluating a Function Rule
Check Understanding #4
Example 5: Finding the Range

Checkpoint Quiz #1
Lessons 5-1 through 5-2
p.246

Homework
p.244 - 245 #3 - 30 x 3's

PreCalculus: Lesson Plan * 10/24/2006

Chapter 3 Exponential and Logarithmic Functions
Section 3.2 Logarithmic Functions and Their Graphs
Objective: Students will know how to recognize, graph and evaluate logarithmic functions.

Quickstart
3.1 Practice
Determine the balance A at the end of 20 years if $1500 is invested at 6.5% interest and the interest is compounded (a) quarterly and (b) continuously.

Lesson
II Graphs of Logarithmic Functions
y = log_ax is the inverse of y = a^x and has the following properties
1. The domain is (0, ∞)
2. The range is (-∞, ∞)
3. The x-intercept is (1, 0)
4. The y-axis, x = 0, is a vertical asymptote
5. It is increasing when (a > 0)

Example 3
Sketch the graph of the following on the same coordinate axis.
a) y = log₁₀x
Vertical asymptote x = 0
x-intercept (1, 0)
Additional point (10, 1)
b) y = log₁₀(x + 2)
Shift 2 units left from part a
Vertical asymptote x = -2
x-intercept (-1, 0)
Additional point (8, 1)
c) y = log₁₀(x + 2) - 1
Shift 2 units left and one unit down from part a
Vertical asymptote x = -2
Additional point (-1, -1)
x-intercept (8, 0)
Use graphing utility to verify results.

III. The Natural Logarithmic Function
The logarithmic function with base e [log_ex] is called the natural logarithmic function and is denoted by ƒ(x) = ln x.

Example 4: Evaluate
a) ln e⁵ = 5
b) e^{ln 3} = 3
c) ln (1 / e²) = ln e^-2 = -2

Example 5: find the domain of the following function.
ƒ(x) = ln (x + 3)
x + 3 > 0 => x > -3
The domain is (-3, ∞)

IV. Application

Example 7
The model
t = 12.5421 ln[x / (x - 1000)], x > 1000
approximates the length of a home mortgage of $150,000 at 8% in terms of the monthly payment. In the model, t is the length of the mortgage in years and x is the monthly payment in dollars. Find the length of the home mortgage of $150,000 at 8% if the monthly payment is $1300.
t = 12.4421 ln(1300 / 300) ≈ 18.4 years or 18 years and 5 months

This picture is from freeimages.co.uk.
It shows a typical American house. Many people use mortgages to purchase homes. The amount of time it takes to pay off a home can be calculated using a logarithmic function.

3.3 Properties of Logarithms
Objectives: Students will know how to rewrite logarithmic functions with a different base, use properties of logarithms to evaluate, rewrite, and expand expressions.
I. Change of Base
Our calculators have only two buttons for logarithmic functions, base 10 [log] and base e [ln]. To evaluate any other logarithmic functions using a calculator, we must rewrite it in one of these bases using the following formula.

Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then
log_ax = log_bx / log_ba = log x / log a = ln x / ln a

Example 1: Evaluate the following
a) log₅18 = ln 18 / ln5 ≈ 1.7959
b) log₂42 = log 42/ log 2 ≈ 5.3923

II. Properties of Logartihms
Logarithms are exponents so the following properties are similar to exponent properties.

Let a be a positive real number such that a ≠ 1, let n be a real number, and let u and v be positive real numbers. Then
1. log_a(uv) = log_au + log_av
2. log_a(u/v) = log_au - log_av
3. log_auⁿ = n log_au

III. Rewriting Logarithmic Expressions

Example 2 Expand the logarithmic expression
a) log(2x³y⁴)
= log2 + logx³ + logy⁴
= log2 + 3logx + 4logy
b) ln [√(x+5) / y²] = ln√(x+5) - ln y² = (1/2) ln(x+5) - 2 ln y

Homework
p.236 -237
# 3 - 45 x 3's, 47 - 52 all, 54 - 60 x 3's, 73

Algebra I: Lesson Plan * 10/23/2006

Quickstart
MA.B.2.4.1 Practice #3 - 7

Lesson
1-9: Graphing Data on the Coordinate Plane Objective: To graph points on the coordinate plane, to analyze data using scatter plots, to interpret graphs in real-world situations

Graphing Points on the Coordinate Plane p.59
New vocabulary: coordinate plane, x-axis, y-axis, origin, quadrants, ordered pair, coordinates, x-coordinate, y-coordinate


Example 1: Identifying Coordinates p.60
Example 2: Graphing Points p.60
Check Understanding #1, 2
In which quadrant would you find each point from Check Understanding #2?

Analyzing Data Using Scatter Plots
Example 4: Making a Scatter Plot p.61
What correlation does the data in Example 4 show, positive, negative or none?

Chapter 5: Graphs and Functions 5-1 Relating Graphs to Events Objective: To interpret, analyze, and sketch graphs from real-world situations.
Chris's Adventure: Sketching a graph to model a real life situation.
Chris got up from his seat, walked slowly to the pencil sharpener, stopped and sharpened his pencil, walked slowly to the trash can, stopped to look out the window, the walked quickly back to his seat when Ms. Lynch called his name.

Interpreting, Sketching, and Analyzing Graphs p.236 - 237
Example 1: Interpreting Graphs
Check Understanding #1
Example 2: Sketching a graph
Check Understanding #2

Assignment
Green Workbook Section 1-9
Green Workbook Section 5-1

PreCalculus: Lesson Plan * 10/23/2006

Chapter 3: Exponential and Logarithmic Functions
3.1 Exponential Functions and their Graphs Objective: Students will be able to recognize, evaluate, and graph exponential functions.

IV. Compound Interest
A. Formulas for Compound Interest [p.222]
B. Finding the Balance for Compound Interest [p.223]

V. Other Applications
A. Population Growth [p.224]

This map is from the US census bureau website
www.census.gov/population/estimates/state/
The census allows the federal government to create models to predict changes in population.

Homework p. 227 #73, 74, 75 (algebraically)