The basics of graphing f(x) = Asin(bx+c) + D

  -->

Here is f(x) = sinx from x = -6 to 6.

Height (amplitude) = 1 (from 1 to -1)

Period = 2pi

f(0) = 0

f(pi/2) =1

f(-pi/2) = -1

Here is f(x) = 2sinx from x = -6 to 6.

Height (amplitude) = 2 (from 2 to -2)

Period = 2pi

f(0) = 0

f(pi/2) =2

f(-pi/2) = -2

The thing that is changed is the height or amplitude. f(x) = Asinx oscillates from A to -A. Choosing a negative A makes the graph start at 0 and move downward instead of upward.

  -->

Here is f(x) = sin2x from x = -6 to 6.

Height (amplitude) = 1 (from 1 to -1)

Period = pi

f(0) = 0

f(pi/4) =1

f(-pi/4) = -1

The thing that is changed is the period. f(x) = sinbx has period 2pi/b.

To repeat faster than 2pi, choose |b| > 1. For slower repeats |b| < 1. Choosing a negative b makes the graph start at 0 and move downward instead of upward.

Here is f(x) = sin(x + pi/4) from x = -6 to 6.

Height (amplitude) = 1 (from 1 to -1)

Period = 2pi

f(-pi/4) = 0

f(pi/4) =1

f(-3pi/4) = -1
What changes here are the x positions of the midpoint and the maximum and minimum values. f(x) = sin(x + c) reaches 0 at x = -c, the high point at x = pi/2 - c and the low point at x = -pi/2 - c.

Here is f(x) = sinx + pi/4 from x = -6 to 6.

Height (amplitude) = 1 (from 1+pi/4 to -1+pi/4)

Period = 2pi

f(0) =pi/4

f(pi/2) =1+pi/4

f(-pi/2) = -1+pi/4

What changes here are the y positions of the midpoint and the maximum and minimum values. Instead of oscillating between 1 and -1, f(x) = sinx + D goes back and forth between D+1 to D-1.


-->

To review.

f(x) = Asin(bx + c) + D

The constant A changes the height or amplitude of the graph, rising to |A| above the middle y value, which is the constant D, and falling to -|A| below the middle y value. Negative values of A cause the graph to go downward to the right of bx + c = 0 instead of upward.

The constant b changes the distance in x between the consecutive highest points, also known as the period of the function.  The standard period is 2pi, and if |b| ≠ 1, the period changes to 2pi/|b|. The constant c changes where on the x-axis the sine function reaches its middle value. This is called phase shift. The point of the middle value moves to bx + c  = 0, so positive values cause the graph to shift left and negative values cause the graph to shift right.
 

Practice for graphing sine waves.

If you want to visualize the graphs you are asked to draw, the website Wolfram|Alpha has an instruction you can type called "plot". Here is the example of plot 3sin(2x/3).

Here are some examples for practice without graphs. You can do that yourself on Wolfram|Alpha. The answers are in the comments.

a] f(x) = 2 + 4sin(x/2)

period = ______

amplitude = _______

maximum of f is ______, reached when x = _______.


minimum of f is ______, reached when x = _______.

b] k(x) = 4 + 2cos(3x)

period = ______

amplitude = _______

maximum of k is ______, reached when x = _______.

minimum of k is ______, reached when x = _______.

c] q(x) = 1 - 3sin(5x)

period = ______

amplitude = _______

maximum of q is ______, reached when x = _______.

minimum of q is ______, reached when x = _______.

Answers in the comments.

Phase shift

We have dealt with three ways to change a trig function.

1) Amplitude
f(x) = asinx or g(x) = acosx

This changes the height of the graph.  Instead of oscillating between -1 and 1, the high and low points will be |a| and -|a|.  (It could be that a is negative, so we need the absolute value signs.)

2) Frequency (or period)

f(x) = sinpx or g(x) = cospx

This changes the distance between the high points.  For sine and cosine, the period is 2pi.  This changes to |2pi/p|.  If |p| > 1, it takes a shorter distance to repeat.  If |p| < 1, the wave is stretched and takes a longer distance to repeat.

3) Adding a constant
f(x) = c + sinx or g(x) = c + cosx

This makes a graph fluctuates between c - 1 and c + 1.  If there is an amplitude multiplier a, the fluctuation is between c - |a| and c + |a|.

The last way to change a trig function is called phase shift.

f(x) = sin(px+h) or g(x) = cos(px+h)

The "important" points in a trig function happen at multiples of ½pi.

sinx = 0 when x = ...-2pi, -pi, 0, pi, 2pi...
sinx = 1 when x = ... -3½pi, -1½pi, ½pi, 2½pi, 4½pi....
sinx = -1 when x = ... -2½pi, -½pi, 1½pi, 3½pi, 5½pi....


cosx = 0 when x = ...-1½pi, -½pi, ½pi, 1½pi...
cosx = 1 when x = ... -2pi, 0, 2pi, 4pi....

cosx = -1 when x = ... -3pi, -pi, pi, 3pi, 5pi....


If we want to look at multiples of  ½pi, we let k be any integer and consider kpi/2.

px+h = kpi/2
px = kpi/2 - h
x = (kpi/2 - h)/p

These will be the values of x where the graph will be either in the middle or at the maximum or minimum.  The easiest thing to do is plug in k = 0, k = 1 and k = 2 in a row to see what you get. That means


x = - h/p
x = (pi/2 - h)/p
x = (pi - h)/p

There are four possible situations you will get plugging these numbers in.

Middle, high, middle
High, middle, low
Middle, low, middle
Low, middle, high

Once you have three such points, you can draw the entire graph.

For the following functions, find three points in a row where the function either reaches an extreme value or a median value.

f(x) = -3 + 2cos(pi(x + ¾))
g(x) = -4sin(2x - 5)
k(x) = cos(2 - 3x)

Answers in the notes.



 





 

Syllabus for Math 50 for Fall 2011

Math 50: Trigonometry Fall 2011 – Laney College
Instructor: Matthew Hubbard
MWF: 9:00-9:50 am
Email address: mhubbard@peralta.edu
Recommended Texts: Trigonometry (open source) http://mecmath.net/trig/trigbook.pdf
Barnett et al, "Analytic Trigonometry with Applications" (10th Edition)
Office hours: MWF: 8:30-8:55 am G-210 (Classroom)

Add and drop class dates
Last date to add: Sat., Sept. 3
Last date to drop class without a “W”: Sat., Sept 17
Last date to drop class with a “W”: Wed., Nov. 23

Holiday schedule for MWF schedule
Labor Day: Monday, Sept. 5
Veteran’s Day: Friday, Nov. 11
Day after Thanksgiving Friday, Nov. 25

Test dates:
Midterm 1: Friday, Sept. 23
Midterm 2: Friday, Oct. 28
Comprensive Final: Wednesday, Dec 14 8:00-10:00 am

Homework to be turned in: Assigned last class period of the week, due next class.
Late homework accepted AT THE BEGINNING of the class after it was due

Quizzes: One on the last class of the week in weeks without a midterm or final

Grading system
Homework 20%
Lab 5%
Midterm I 25%*
Midterm II 25%*
Quizzes 25%*
Final 25%

Lowest two of the homework scores will be dropped from the total.
Lowest two of the quiz scores will be dropped from the total.
*Lowest total out of 100 points the quiz total and two midterms will be dropped from the final grade.
Anyone getting a higher grade out of 100 points on the final than the weighted average of all grades combined will get the final percentage instead deciding the final grade.
This option is only available to students who have missed at most three homework assignments.
The extra 5%: Extra credit will be factored into quizzes, midterms and homework at about 5%. This means 100 point tests with have 105 points and homework and quizzes will average 21 points each, though they will count as 20 points.
Class rules:
All cell phones and electronic communication devices off during class.
No hats, hoodies or headphones worn during quizzes and exams.
No calculators that also combine a cell phone or text message machine.

Recommended calculator: TI-30XIIs (any calculator with at least two lines of output will do, the TI-30XIIs is the cheapest that does all the things you need to do in this class. If you need help with any Texas Instruments calculator, I should be able to steer you in the right direction. I haven’t used other brands of calculators as much.)

Open source textbook: The textbook is free and online. It helps a lot if you have a computer and Internet at home so you can have your own copy on your computer. You will need the Adobe Reader software, which is free and can be downloaded from several websites.

Academic honesty: All assignments you turn in, homework, exams and quizzes, must be your own work. Anyone caught cheating on these assignments will be punished, where the punishment can be as severe as failing the class or being put on college wide academic probation.

Student learning outcomes
Math 50 Trigonometry
• Evaluate the 6 trigonometric functions using a calculator, as well as determining exact values for some special angles without a calculator.
• Solve a triangle (right, acute, obtuse), given various angles and sides.
• Convert between decimal degrees, degree-minute-seconds, and radian measure of an angle.
• Demonstrate knowledge of several trigonometric identities and use them to verify other identities.
• Graph trigonometric functions.
• Solve trigonometric equations.

Syllabus for Fall 2012

Math 50: Trigonometry (L-41147)            Fall 2012 – Laney College
Instructor:  Matthew Hubbard                TTh: 10:30-11:45 am
Email address: mhubbard@peralta.edu       
Recommended Text: Trigonometry (open source) http://mecmath.net/trig/trigbook.pdf
Class website: http://thetrig.blogspot.com/

Office hours:     MW: 8:30-8:55 am F-203 (Classroom)
        T-Th: 10:00-10:25 am G-206 (Classroom)

Add and drop class dates
Last date to add:                    Sun., Sept. 2
Last date to drop class without a “W”:        Sat., Sept. 1
Last date to drop class with a “W”:            Sat., Nov. 17

Holiday schedule for T-Th schedule
Thanksgiving                        Thursday, Nov. 22

Test dates:
Midterm 1:        Thursday, Sept. 27
Midterm 2:        Thursday, Nov. 1
Comprehensive Final:    Thursday, Dec. 13 10:00 am- noon

    Homework to be turned in:  Assigned last class period of the week, due next class.
    Late homework accepted AT THE BEGINNING of the class after it was due
    Quizzes:    One on the last class of the week in weeks without a midterm or final

        Grading system

Homework            20%
Lab                  5%
Quizzes            25%
Midterm I            25%
Midterm II            25%
Final                25%
Voting bonus              2%   
 If you are eligible to vote, bring an “I voted” sticker
                    If you are too young to vote, bring proof of age
                    If you are not a U.S. citizen, bring proof of status 

Lowest two of the homework scores will be dropped from the total.
Lowest two of the quiz scores will be dropped from the total.
Lowest total out of 100 points the quiz total and two midterms will be dropped from the final grade.
Anyone getting a higher grade out of 100 points on the final than the weighted average of all grades combined will get the final percentage instead deciding the final grade.  This option is only available to students who have missed at most two homework assignments.
Anyone going into final with 97.5% class average can skip the final and be awarded an A.


Class rules:  All cell phones and electronic communication devices off during class.
No hats, hoodies or headphones worn during quizzes and exams.
No calculators that also combine a cell phone or text message machine.

Recommended calculator: TI-30XIIs (any calculator with at least two lines of output will do, the TI-30XIIs is the cheapest that does all the things you need to do in this class.  If you need help with any Texas Instruments calculator, I should be able to steer you in the right direction.  I haven’t used other brands of calculators as much.)

Open source textbook:  The textbook is free and online.  It helps a lot if you have a computer and Internet at home so you can have your own copy on your computer.  You will need the Adobe Reader software, which is free and can be downloaded from several websites.  I will be assigning problems from the book occasionally.

Academic honesty:  All assignments you turn in, homework, exams and quizzes, must be your own work.  Anyone caught cheating on these assignments will be punished, where the punishment can be as severe as failing the class or being put on college wide academic probation.

Students with disabilities
The Disabled Students Program Services (DSPS) should have your academic accommodation with the instructor.  After the first day, I will accept these accommodations electronically or by hard copy on paper.  If you need academic accommodation and have not yet applied, please call 510-464-3428 for an appointment.

Exam policies
Tests will be closed book and closed notes. Some information you will be expected to remember, other formulas and information will be provided. No sharing of calculators is allowed.  You are responsible for knowing how to use your calculator to find answers.

            Student learning outcomes

Math 50 Trigonometry
•    Evaluate the 6 trigonometric functions using a calculator, as well as determining exact values for some special angles without a calculator.
•    Solve a triangle (right, acute, obtuse), given various angles and sides.
•    Convert between decimal degrees, degree-minute-seconds, and radian measure of an angle.
•    Demonstrate knowledge of several trigonometric identities and use them to verify other identities.
•    Graph trigonometric functions.
•    Solve trigonometric equations.

The reciprocal relationship

The teacher will be on time and prepared to teach the class.
The students will be on time and prepared to learn.

The teacher will present the material to the best of his ability.
The students will absorb the material to the best of their ability.  They will ask questions when topics are not clear.

The teacher will do his best to answer the questions the students ask about the material, either by repeating an answer with more details included or by taking a different approach to the material that might be clearer to some students.
The students will understand if the teacher feels a topic has been covered enough for the majority of the class and will accept questions being answered outside the class, either in extra time or through written communication.

The teacher will do his best to keep the class about the material.  Personal details and distractions that are not germane to the class should not be part of the class.
The students will do their best to keep the class about the material.  Questions that are not about the topic should be avoided.  Distractions like cell phones and texting are not welcome when the class is in session.

The teacher will give assignments that will help the students master the skills required to pass the course.
The students will put in their best efforts to complete the assignments.
When the assignments are completed, the teacher will make every effort to get the assignments graded and back to the students in a timely manner, by the next class session whenever possible.

The teacher will present real life situations where the skills being learned will be used when they exist.  In math, sometimes a particular skill is needed in general to solve later problems that will have real life applications.  Other skills have the application of “learning how to learn”, of committing an idea to memory so that committing other ideas to memory becomes easier in the long run.
The student has the right to ask “When will I use this?” when dealing with mathematical topics.  Sometimes, the answer is “We need this skill for the next skill we will learn.”  Other times, the answer is “We are learning how to learn.”  Both of these answers are as valid in their way as “We will need this to understand perspective” or “We use this to balance our checkbooks” or “Ratios can be used to figure out costs” or other real life applications.