Hi everyone,
Lesson 3.6 concludes the material that will be covered on the 3.1 - 3.6 Test (The test is Friday, Sept. 19) In this lesson, we solved rational inequalities. The solution process utilizes the following steps:
Find the zeros and the excluded domain values. These numbers divide the real number line into intervals. Pick a "test point" in each interval and test it in the original function. If the "test point" produces a true inequality, then the interval is in the solution set. If the "test point" produces a false inequality, then the interval is not in the solution set.
This lesson uses the term "strict" for an inequality such as < or >.
I want each of you to know that I am pleased with your progress in this class. I realize that the transition into a Pre AP class has been difficult for many of you. You are a hard working group and I commend you :)
Mrs. S
Wednesday, September 17, 2008
Monday, September 15, 2008
TEST BONUS OPPORTUNITY: Check out the link to Plus magazine
How long will you live? Who will win the Presidential election? How should you write down numbers? Who's your ideal partner? How good is our voting system? What is a differential equation? These are difficult and momentous questions. This issue of Plus has some answers, along with a tour of digital art and the usual range of podcasts, news and reviews.
Plus is an online magazine that has some really neat stuff! See how you can use this site to
a. Find real life stuff that uses the math we are learning.
b. Find neat ideas and connections that you've never heard of before
Bonus opportunity: Read any issue of Plus Magazine. Create a 5 minute power point presentation telling a story about what you learned. (Your power point should just be an outline...most of the story will be told by you.) Present your story to the class and receive from 1 to 5 bonus points on a test.
Plus is an online magazine that has some really neat stuff! See how you can use this site to
a. Find real life stuff that uses the math we are learning.
b. Find neat ideas and connections that you've never heard of before
Bonus opportunity: Read any issue of Plus Magazine. Create a 5 minute power point presentation telling a story about what you learned. (Your power point should just be an outline...most of the story will be told by you.) Present your story to the class and receive from 1 to 5 bonus points on a test.
Lesson 8.6 Adv. Alg./Trig
Natural logs are base e logarithms. Example: ln 3
All log properties for common logs apply to natural logs.
* e is approximately equal to the irrational number 2.71828182845904523....
* To write a natural log as a single logarithm, use the product,
quotient, and power rules.
* To solve a natural log equation, simplify then change to
exponential form and solve for the variable.
* To solve an exponential equation in base e, simplify,
take the natural log of both sides of the equation, use the
power rule, then solve for the variable.
Tomorrow we will work the application problems in 8.6 and review
8.5 and 8.6 by working group problems.
All log properties for common logs apply to natural logs.
* e is approximately equal to the irrational number 2.71828182845904523....
* To write a natural log as a single logarithm, use the product,
quotient, and power rules.
* To solve a natural log equation, simplify then change to
exponential form and solve for the variable.
* To solve an exponential equation in base e, simplify,
take the natural log of both sides of the equation, use the
power rule, then solve for the variable.
Tomorrow we will work the application problems in 8.6 and review
8.5 and 8.6 by working group problems.
Sunday, September 14, 2008
PAPPC Lesson 3.5
Lesson 3.5 is all about graphing rational functions. Make sure that you know the difference between a rational function and a polynomial function. Here are the hi-lites of the lesson: Before sketching the graph, find the Domain, Hole(s), Vertical Asymptotes, Horizontal or Oblique Asymptotes, and Zeros and y-intercept. After graphing, determine the range of the function.
Graphing Tips: 1. Sketch hole(s), zero(s), and y-intercept of the graph.
2. Sketch vertical, horizontal or oblique asymptote(s)
3. Note if the ratio of the leading coefficients is
positive or negative.
a. If positive, the graph begins on the right of and above
the x-axis.
b. If negative, the graph begins on the right of and below
the x-axis.
4. Sketch the first branch of the graph in the appropriate region.
5. Use the multiplicity of the vertical asymptote to determine from
which infininty to begin the next branch of the graph.
6. Repeat steps 4 and 5 until the graph is complete.
Please help one another by commenting with your own tips and helps for graphing. See you Monday.
Mrs. S
Graphing Tips: 1. Sketch hole(s), zero(s), and y-intercept of the graph.
2. Sketch vertical, horizontal or oblique asymptote(s)
3. Note if the ratio of the leading coefficients is
positive or negative.
a. If positive, the graph begins on the right of and above
the x-axis.
b. If negative, the graph begins on the right of and below
the x-axis.
4. Sketch the first branch of the graph in the appropriate region.
5. Use the multiplicity of the vertical asymptote to determine from
which infininty to begin the next branch of the graph.
6. Repeat steps 4 and 5 until the graph is complete.
Please help one another by commenting with your own tips and helps for graphing. See you Monday.
Mrs. S
Lesson 8.5 Adv. Alg./Trig
Lesson 8.5
This lesson has 3 objectives. On Friday, Sept. 12, we learned how to solve exponential equations using four easy steps. You have a copy of these steps on your "footprints" graphic organizer. We also learned the second objective which is to solve logarithmic equations using two easy steps which are also on your "footprints" graphic organizer.
On Monday, Sept. 15, we will learn how to use the "change of base formula" for logarithms. This is the third and final objective of lesson 8.5
This lesson has 3 objectives. On Friday, Sept. 12, we learned how to solve exponential equations using four easy steps. You have a copy of these steps on your "footprints" graphic organizer. We also learned the second objective which is to solve logarithmic equations using two easy steps which are also on your "footprints" graphic organizer.
On Monday, Sept. 15, we will learn how to use the "change of base formula" for logarithms. This is the third and final objective of lesson 8.5
Wednesday, September 10, 2008
Advanced Algebra/Trig Lesson 8.4
Lesson 8.4 is all about the three logarithm properties.
The product rule changes the product of a single log to the sum of two logs.
The quotient rule changes the quotient of a single log to the difference of two logs.
The power rule allows us to move an exponent down and multiple it times the log of the given number.
Bonus Question: You will receive a bonus point on the next quiz if you e-mail me the correct answer before 8 am on Thurs., Sept. 11, 2008
Write (logX - logY + logZ) as a single logarithm.
e-mail your answer to tsnipes@mscs.k12.al.us. Do not post your answer as a comment to this posting.
The product rule changes the product of a single log to the sum of two logs.
The quotient rule changes the quotient of a single log to the difference of two logs.
The power rule allows us to move an exponent down and multiple it times the log of the given number.
Bonus Question: You will receive a bonus point on the next quiz if you e-mail me the correct answer before 8 am on Thurs., Sept. 11, 2008
Write (logX - logY + logZ) as a single logarithm.
e-mail your answer to tsnipes@mscs.k12.al.us. Do not post your answer as a comment to this posting.
Tuesday, September 9, 2008
PAPPC LESSON 3.4
Your task today is to find the domain of a rational function.
You must also know how to find the vertical, horizontal, and slant asymptotes of a rational function.
To determine the domain, you must find the values that cause the denominator to = zero. These values are NOT in the domain. It may help you to "see" the domain if you sketch a number line, put open circles for undefined values, and shade the line for values that are in the domain. Use your number line as an aid in writing the interval(s) that make up the domain.
Use you class notes to study vertical, horiz., and slant asymptotes. If you have any questions, please ask for help online.
Bonus Question: Worth 2 point on Friday's quiz. e-mail your answer to tsnipes@mscs.k12.al.us Your e-mail must be posted before 8 am on Thurs. Sept. 11, 2008
1. Explain how to find the holes and vertical asymptotes of a rational function.
2. Explain why a polynomial function will never have any vertical asymptotes or
holes."
Mrs. S
You must also know how to find the vertical, horizontal, and slant asymptotes of a rational function.
To determine the domain, you must find the values that cause the denominator to = zero. These values are NOT in the domain. It may help you to "see" the domain if you sketch a number line, put open circles for undefined values, and shade the line for values that are in the domain. Use your number line as an aid in writing the interval(s) that make up the domain.
Use you class notes to study vertical, horiz., and slant asymptotes. If you have any questions, please ask for help online.
Bonus Question: Worth 2 point on Friday's quiz. e-mail your answer to tsnipes@mscs.k12.al.us Your e-mail must be posted before 8 am on Thurs. Sept. 11, 2008
1. Explain how to find the holes and vertical asymptotes of a rational function.
2. Explain why a polynomial function will never have any vertical asymptotes or
holes."
Mrs. S
Sunday, September 7, 2008
Advanced Algebra/Trig Lesson 8.3
Logarithmic functions as Inverses
You may wonder about the title of this lesson and what it means. The logarithmic function (log function) has an inverse relationship with the exponential function we studied in lesson 8.1
Think about the graph for y = 2^x.
Now, look on page 440 of your text and look at the graph under objective 2.
Notice the exponential funtion is sketched in red ink and the logarithmic function is sketched in blue ink.
Now look at the graph in example 5. What other equation has been graphed in green?
In Alg. 2, you learned that a function and its inverse are "SYMMETRIC" about the line y = x. This means that if you fold the graph on the line y = x, the exponential function and the log function fold on top of each other.
ISN'T MATH NEAT... Again, looking at the graph in example 5, look
at the asymptote for y = log (base 2) of x.
The asymptote for the logarithmic function is the y axis or the line x = 0.
As you remember, the asymptote for the exponential function is the line y = 0.
Tonight's assignment: Post a comment and answer the following QUESTIONS?????????????????????????????????????????
1. In this lesson, what did you learn about a common log?
2. How do you change an exponential equation (y=b^x) to a log equation?
3. How do you change a log equation to an exponential equation?
You may wonder about the title of this lesson and what it means. The logarithmic function (log function) has an inverse relationship with the exponential function we studied in lesson 8.1
Think about the graph for y = 2^x.
Now, look on page 440 of your text and look at the graph under objective 2.
Notice the exponential funtion is sketched in red ink and the logarithmic function is sketched in blue ink.
Now look at the graph in example 5. What other equation has been graphed in green?
In Alg. 2, you learned that a function and its inverse are "SYMMETRIC" about the line y = x. This means that if you fold the graph on the line y = x, the exponential function and the log function fold on top of each other.
ISN'T MATH NEAT... Again, looking at the graph in example 5, look
at the asymptote for y = log (base 2) of x.
The asymptote for the logarithmic function is the y axis or the line x = 0.
As you remember, the asymptote for the exponential function is the line y = 0.
Tonight's assignment: Post a comment and answer the following QUESTIONS?????????????????????????????????????????
1. In this lesson, what did you learn about a common log?
2. How do you change an exponential equation (y=b^x) to a log equation?
3. How do you change a log equation to an exponential equation?
Friday, September 5, 2008
Lesson 3.3 PAPPC
3.3 POLYNOMIAL FUNCTIONS AND MODELS
This lesson has THREE BIG OBJECTIVES :)
A. Identify polynomial functions and their degrees (See ex. 1 page 201
B. Identify the zeros of a polynomial function and their multiplicity. Know how to obtain the degree of a function by adding the multiplicities of the function.
Know when the graph crosses the x-axis and when the graph touches(is tangent to) the x-axis. See pages 204-205
C. analyze the graph of a polynomial function. Know the 4 "End Behavior" rules See pages 206 - 208. Work examples 5 and 6.
I will be online tonight(Sept. 8) from 7 to 9 pm. Leave me a question, comment, or tell me something GOOD that you have learned in lesson 3.3!
This lesson has THREE BIG OBJECTIVES :)
A. Identify polynomial functions and their degrees (See ex. 1 page 201
B. Identify the zeros of a polynomial function and their multiplicity. Know how to obtain the degree of a function by adding the multiplicities of the function.
Know when the graph crosses the x-axis and when the graph touches(is tangent to) the x-axis. See pages 204-205
C. analyze the graph of a polynomial function. Know the 4 "End Behavior" rules See pages 206 - 208. Work examples 5 and 6.
I will be online tonight(Sept. 8) from 7 to 9 pm. Leave me a question, comment, or tell me something GOOD that you have learned in lesson 3.3!
Lesson 3.2 PAPPC
POWER FUNTIONS AND MODELS
Lesson 3.2 is all about the properties of power functions of even and odd degrees. The most important concept for 3.2 is this: You should be able to sketch the graph of a power function, by looking at its degree and determining the end behavior of the graph.
The properties of power functions are discussed on pages 196 and 197. Here is an example: y = x^4 flattens at the vertex and gets more vertical as -1 < x < 1.
Read about these properties, and play with graphs on your TI-84 to learn more.
TI-84 exercises:
Graph y = x^2, y = x^2 + 1, and y = (x - 4)^2 + 1. What do you notice about the "end behavior" of all graphs. Now graph y = (x-1)(x-4). How is this graph different from the first three?
Leave me a comment with your observations and your questions on 3.2 :)
Lesson 3.2 is all about the properties of power functions of even and odd degrees. The most important concept for 3.2 is this: You should be able to sketch the graph of a power function, by looking at its degree and determining the end behavior of the graph.
The properties of power functions are discussed on pages 196 and 197. Here is an example: y = x^4 flattens at the vertex and gets more vertical as -1 < x < 1.
Read about these properties, and play with graphs on your TI-84 to learn more.
TI-84 exercises:
Graph y = x^2, y = x^2 + 1, and y = (x - 4)^2 + 1. What do you notice about the "end behavior" of all graphs. Now graph y = (x-1)(x-4). How is this graph different from the first three?
Leave me a comment with your observations and your questions on 3.2 :)
Lesson 3.1 PAPPC
Many real life applications are modeled by quadratic functions. Think about some of the application problems in this lesson. What real life applications did you model using a quadratic function? It is important that you can name some of these applications. In particular, know that a revenue equation = xp where x is the number of goods sold and p is the price.
Other important objectives from 3.1 are.....
1. Write quadratic equations in quadratic form, or in vertex form, when given the graph with vertex and y-intercept.
2. Graph quadratic functions by hand by determining and labeling the vertex, axis of symmetry, y-intercept, vertical compression/stretch.
3. Given an equation in quadratic form, know the equation of the axis of symmetry, and know at first glance the y-intercept.
4. Know how the value of a determines if there is a vertical stretch, or a vertical compression.
5. EXTRA EXTRA EXTRA!! (required info that is not in our text)
The focus of the parabola (in vertex form) is k + 1/(4a) OR K - 1/(4a)
The LATUS RECTUM = 1/a (THE L.R. IS the line passing thru the focus.)
Please leave a comment if you have questions re. problems in this lesson.
Mrs. S
Other important objectives from 3.1 are.....
1. Write quadratic equations in quadratic form, or in vertex form, when given the graph with vertex and y-intercept.
2. Graph quadratic functions by hand by determining and labeling the vertex, axis of symmetry, y-intercept, vertical compression/stretch.
3. Given an equation in quadratic form, know the equation of the axis of symmetry, and know at first glance the y-intercept.
4. Know how the value of a determines if there is a vertical stretch, or a vertical compression.
5. EXTRA EXTRA EXTRA!! (required info that is not in our text)
The focus of the parabola (in vertex form) is k + 1/(4a) OR K - 1/(4a)
The LATUS RECTUM = 1/a (THE L.R. IS the line passing thru the focus.)
Please leave a comment if you have questions re. problems in this lesson.
Mrs. S
Adv. Alg./Trig Lesson 8.1
Advanced Algebra Trig Lesson 8.1 EXPONENTIAL FUNCTIONS
This lesson introduces us to the exponential function: y = ab^x.
If b > 1, the equation represents exponential growth.
if 0 < b < 1, the equation represents exponential decay (also known as depreciation)
In this equation, b is called the "growth factor", or the "decay factor". In real world applications b = 1 + r (where r is the rate as a decimal). In depreciation problems, r is negative. In growth problems, r is positive.
a represents the y-intercept of the graph. (0,a).
draft 1/24/09 by Mrs. S.
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