Algebra II Chapter 7 Notes

Name:______Period:______

Day 1: Graph Exponential and growth functions:

Exponential Function: where and the base b is a positive number other that 1. / Exponential Growth Function:
where b > 1.
Parent function for exponential growth:
/ Growth factor = b
Asymptote: a line that a graph approaches more and more closely.
Domain: x values for the function
Range: y or f(x)values for the function
  1. Find five values of the function
  2. Plot the points on the graph
  3. Connect the points with a smooth curve from left to right. The shape is similar to the parent function graphed above
  4. Identify the domain and range of the graph.

Example #1 Graph

X / Y

Domain:

Range:

Example #2: Graph

X / Y

Domain:

Range:

Example #3:

1.

2. Graph

3. Translate graph h units horizontally and k units

vertically for each point on the graph.

X / Y

Domain:

Range:

Day 2: Graph Exponential Decay Functions:

Exponential Decay functions: where 0 < b > 1
Parent function for Exponential Decay:

Base (b) is called the decay factor.
  1. Find five values of the function
  2. Plot the points on the graph
  3. Connect the points with a smooth curve from left to right. The shape is similar to the parent function graphed above
  4. Identify the domain and range of the graph.

Example #1:

X / Y

Domain:

Range:

Example #2:

Graph

X / Y

Domain:

Range:

Example # 3:

Graph

1.

2. Graph

3. Translate graph h units horizontally and k units

vertically for each point on the graph.

X / Y

Domain:

Range:

Day # 3: Use functions involving e

The natural base: e (an

irrational Number) 2.718281828

Use it like a variable:

1. 2. 3.

Use a calculator to evaluate:

4. 5.

Natural Base Functions /
  • If a > 0 and r > 0 it is exponential growth function
/
  • If a > 0 and r < 0 then it is exponential decay function

6. Graph

  • Determine if it is growth or decay
  • Plug in 0 and 1 for x and plot those points.
  • Draw a curve connecting those points
  • Determine domain and range

X / Y
0

Domain:______

Range:______

7. Graph:

  • Determine if it is growth or decay
  • Plug in 0 and 1 for x and plot those points.
  • Draw a curve connecting those points
  • Determine domain and range
X / Y
2 /
-1

Domain:______

Range:______

Day4: Evaluate Logarithms and graph Logarithmic Functions

x = ____

x = ____

x = ____

x = ____

This is why we use logs.

if and only if /
To rewrite log into exponent form:
1. Base of log(b) becomes base of exponent
2. Answer(x) becomes exponent
3. Logarithm(y) becomes answer

Logarithmic Form / Exponential Form

Logarithm of 1: regardless of base it always = 0. because

Logarithm of b with base b: always equals 1. because

(If the base and logarithm match the answer is 1.)

Evaluate the logarithm:

Set it equal to “x” and change to exponential form.

1. 2. 3.

Common Logs: Have a base ___ and can be done in a calculator. If the problem has no base it is understood to be base ____.

Natural Logs: Have a base ___. It can be denoted as or.

Evaluate the logarithm (use a calculator if necessary):

4. 5. 6. 7.

8. 9.

HINT: rewrite so base of log and logarithm are the same, then answer is the exponent.

Day # 5 Review 7.2 - 7.4

Graph the function. State the domain and range.

1.

X / Y

Domain:______

Range:______

Graph the function and state the domain and range.

2.

X / Y
0
-1 /

Domain:______

Range:______

Evaluate the logarithm without using a calculator:

3. 4.

5. 6.

Day # 6: Apply Properties of Logarithms

Product Property /
Quotient Property /
Power property /

Using properties to evaluate logarithms :( Useand)

1. 2. 3.

Expand a logarithmic expression :( use properties above)

4. Expand 5.

4a. Condense: 5a. Condense:

Change of base: to evaluate any logarithm using a calculator

=

Natural and common logs are in the calculator.

Practice with a calculators:

6. 7.

Day # 7: Solve Exponential and Logarithmic Equations

Rewrite bases to same base and set exponents equal

Use all that you have learned about logarithms and exponential equations to solve for x.

Ex.1:

1. Rewrite 4 and as powers with base 2. and

2. If the logarithms are the same then the exponents are equal so:

3. Solve for x.

Practice:

1. 2.

Taking Log of both sides of an equation:

If the logarithms and bases are not the same and cannot be re-written as the same then take the log of both sides and use change of base to solve.

Ex. 2:

Take log base 4 of each side. If the base of the log is the same as the log then they cancel each other out and you are left with the exponent.

Practice:

3. 4. 5.

If base of logs is the same then logs are equal.

For solving logarithmic equations get the bases of the logs the same.

Ex.3:

Since both bases (5) are the same then:

Solve for x.

Practice:

6.

Exponentiating to Solve Equations

If you do not have logs with same base then exponentiate each side to a base that matches the base of the log so they cancel.

Ex.4:

1. Exponentiate each side of the equation to base 4 to match the base of the log.

2. Since then (5x-1) =

3. Solve for x.

Practice;

7.

Day 7: Exponential decay and growth models

Exponential Growth models-When real-life quantity increases by a fixed percent each year or other time period. /
a = initial amount
1 + r = growth factor for this situation
t = time period /

Ex: 1 In 1996, there were 2573 computer viruses and other computer security incidents. During the next 7 years, the number of incidents increased by about 92% each year.

Write an exponential growth model giving the number (n) of incidents t years after 1996. About how many were there in 2003?

Compound Interest- Interest paid on initial investment, called principal, and on previously earned interest. /
A = amount in account after t years of interest
t = years of interest
r = annual rate of return( expressed as a decimal)
n = number of times compounded in a year
annually = 1 time
semi annually = 2 times
quarterly = 4 times
monthly = 12 times

Ex. #2: You deposit $4000 in an account that pays 2.92% annual interest. Find the balance after 1 year if the interest is compounded:

A: quarterly

B: daily:

Exponential decay models: When a real life quantity decreases by a fixed percent each year(or other time period), the amount y of the quantity after t years. /
a = initial amount
1 – r = decay factor
t = time period /

Ex. #3: A new Honda costs $ 24,000. The value of the car decreases every year (depreciation) by 10% each year. Write the exponential decay model giving the car’s value after t years. Estimate the value after 3 years.

Continuously Compounded Interest /
A = amount in account after compounding
P = Principal( beginning amount)
r = is annual interest rate
t = amount of years /

Ex.#4 You deposit $4000 in an account that pays 6% annual interest compounded continuously. What is the balance after 1 year?

1