swbat - st. joseph high school

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Saint Joseph's High SchoolPrecalculus Guided NotesSemester 1 2009/2010

Instructor: Mr. Brian Baumer

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SWBAT

Unit 1: Functions and Mathematical Models

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Background/ExplanationExamplesVocabulary

y = 20 + 70(0.8)x

4 Representations:Graphically

Numerically

Algebraically

VerballyThe figure shows the temperature, y, as a function of time, x. At x = 0 the coffee has just been poured. The graph shows that as time goeson, the temperature levels off, until it is so close to room temperature, 20°C,that you can’t tell the difference.

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Mathematical Models: Functions that are used to make predictions and interpretations aboutsomething in the real world.

Dependent variable: The quantity represented on the vertical (y) axis. This quantity depends on the change in the horizontal (x) variable.

Independent Variable: The quantity represented on the horizontal (x) axis. This quantity influences the change in the vertical (y) axis variable.

Example 1:

The time it takes you to get home from a footballgame is related to how fast you drive. Sketch areasonable graph showing how this time and speedare related. Tell the domain and range of thefunction.

Extrapolation: Using a function to estimate a value outside the range of the given data.

Interpolation: Using a function to estimate a value within the range of given data.

Assume a distance of 6 miles.

First things first...

We are measuring 2 quantities and we need to decide which is our independent (x) and which is our dependent (y).

Quantities:

1) Time (minutes)2) Average Speed (mph)

It seems reasonable to model this problem with time depending on our average speed. (Time is a function of speed)

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Relation (p. 8): Any set of ordered pairs.

Function (pp. 3, 8): A relationship between two variablequantities for which there is exactly one value of thedependent variable for each value of the independentvariable in the domain.

Relations and Functions:

Functions is a subset of relations. Every function is a relation, but not every relation is a function.

Functions

Relations

Is a relation a function? Use the VERTICAL LINE TEST

Passes: Is a function

Doesn't Pass: Is NOT a function

The yintercept of a function is the value of y when x = 0. It gives the place where the graph crosses the yaxis.

An xintercept is a value of xfor which y = 0. Functions can have more than one xintercept.

quadratic function f(x) = x2 + 5x + 3

Function Notation:

to substitute 4 for x in thequadratic function f you would write: f(4) = 16 + 5(4) + 3 f(4)= 39

The symbol f(4) is pronounced “f of 4” or sometimes “f at 4.” You mustrecognize that the parentheses mean substitution and not multiplication.

Argument (p. 8): In f (x), the variable x or any expression substituted for x.

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Functions

Relations

Is a relation a functions? Use the VERTICAL LINE TEST





Families of functions

Linear Function

f(x) = mx b

Slope intercept form.

The power of the function is 1 (the exponent is assumed to be 1 (degree of 1)

Quadratic FunctionStandard Formf(x) = ax2 + bx + c

Opens up if a > 0. Has a vertex (turning point)The degree is 2.

Cubic Functionf(x) = ax3...Degree of 3

Exponential Functionsf(x) = abx

Your variable is the exponentb> 1 growth. b < 1 decay

growthdecay

Has Asymptotes

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Functions

Relations

Is a relation a functions? Use the VERTICAL LINE TEST





Families of functionsPower Function

f(x) = axbwhere x is variable, b is constant.b is > 0 called positive

b is < 0 called negativepositivenegative

Domain is ≥0 (includes the origin)

Domain is > 0 Negative has vertical asymptote

f(x) = log (x)

Logarithmic

Inverse Variation

f(x) = k x

Has Asymptotes

Rational Functionf(x) = p(x)

g(x)Wacky graphs with points of discontinuity

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f(x) = –x2 + 5x + 3 in the domain 0 ≤ x ≤ 4. Plot the Graph

Restricting the domain for a function can be done on our calculators by entering our functions as seen below.Without the

resticted domain

With the variables resticted

Relating weight to height as we get older.While this could look like an exponential growth function, the power function is a better fit b/c if we extended the graph we would pass through the origin, which a positive power function does.

Graph with restrictions

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Argument (p. 8): In f (x), the variable x or any expression substituted for x.

Dilation magnification, enlarging or reducing.

Translation: A shifting of a graph. This could be horizontal, vertical, or both which would result in a diagonal translation.

DilationThe equation of the preimage function inFigure 13a is

Consider this function in relation to the preimage

Dilation of the x

Dilation of the y

Multiply each x by the reciprocal of the value

Recognizing dilation in function graphs.

Given: f(x) and YOU think that the x's have been dilated by 5 and the y's by 3, then you can write...

Recognizing dilation in function graphs.

Given: f(x) and YOU think that the x's have been dilated by 5 and the y's by 3, then you can write...

The dilation in x is your new argument and it replaces the original argument of just x.

Preimage

The y values get multiplied by this number

Image

What is the argument?

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The outside/ inside issue is relevant for both dilation and translation.

Outside effects y's Inside effects x's and changes our argument, or input.

Given y = 2 + f(x)

What translation has occured.

Given y = f(x4)

What translation has occured.

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Rules for working with composite functions.

1) Work inside out (or right to left)

2) The output of a function becomes the input (argument) of the next function in the process. Your input is your ARGUMENT

f (g (x)) means that x is actually your input, or argument.

or

a) f(g(3))

b) g(f(3))

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Rules for working with composite functions.

1) Work inside out (or right to left)

2) The output of a function becomes the input (argument) of the next function in the process. Your input is your ARGUMENT

f (g (x)) means that x is actually your input, or argument.

or

c) f(f(3))

d) f(g(x)) algebraically, explicitly.

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b. Try to find f(g(8)). Explain why this value is undefined.

c. Try to find f(g(2)). Explain why this value is undefined even though 2 is in the domains of both f and g.

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Background/ExplanationExamplesVocabularyd. Plot f, g, f of g , and g of f on your grapher.

e. Find an equation that expresses f(g(x)) explicitly in terms of x. Find thedomain and range of f g algebraically and show that they agree with thegraph in part d.

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e. Find an equation that expresses f(g(x)) explicitly in terms of x. Find thedomain and range of f g algebraically and show that they agree with thegraph in part d.

write a compound inequality using the function of your first consideration with the domain of your second consideration.

The domain of the composite function is where this domain overlaps the starting domain of our first function.

10 2 3 4 5 6 7 8 9 1012345678910

Domain

This should agree with the domain from the graph of our composite.

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Finding inverses....

Numerically: Switch the x's and y's in a table of values.

Algebraically: Switch the x and y and solve for y.

What do inverses look like compared to each other?

Given y = 0.5x2 + 2,

a. Write the equation for the inverse relation by interchanging the variables and transforming the equation so that y is in terms of x.

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Finding inverses....

Numerically: Switch the x's and y's in a table of values.

Algebraically: Switch the x and y and solve for y.

What do inverses look like compared to each other?

Given y = 0.5x2 + 2,

b. Plot the function and its inverse on the same screen, using equal scales for the two axes. Explain why the inverse relation is not a function.

c) Describe what inverses look like.

d) The domain and the range of a function and its inverse simply switch.

f 1 is read as f inverse or the inverse of f

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Parametric Equations: equations in which both our x and our y are described in terms of a 3rd variable, known as T

Given y = 0.5x2 + 2, enter it and its inverse using parametric equations.

Switching our calculator to PARAMETRIC MODE.

MODE and choose PAR

In parametric mode your window creates your restrictions.

Invertible: when both a function and its inverse are functions, they are described as being invertible.

Our function is not invertible.

OnetoOne Function: A function that passes the "horizontal line test" and vertical line test. Every y produces only 1 x and every x produces only 1 y.

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Background/ExplanationExamplesVocabularyLet f(x) = 3x + 12.

a. Find the equation of the inverse of f. Plot function f and its inverse on the same screen.

b. Explain how you know that f is an invertible function.

c. Show algebraically that the composition of f 1 with f is f 1(f (x)) = x.

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Background/ExplanationExamplesVocabulary Reflections Across the xaxis and yaxis

Example 1 shows you how to plot the graphs in Figure 16a.

The preimage function y = f(x) in Figure 16a is f(x) = x2 – 8x + 17, where2≤ x ≤5.

a. Write an equation for the reflection of this preimage function across the yaxis.

Reflection across the the yaxis is really just a dilarion by a factor of 1.

b. Write an equation for the reflection of this preimage function across the xaxis.

c. Plot the preimage and the two reflections on the same screen.

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Background/ExplanationExamplesVocabularyReflections Across the xaxis and yaxis

Example 1 shows you how to plot the graphs in Figure 16a.

The preimage function y = f(x) in Figure 16a is f(x) = x2 – 8x + 17, where2≤ x ≤5.

b. Write an equation for the reflection of this preimage function across the xaxis.

c. Plot the preimage and the two reflections on the same screen.

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SWBAT

Unit 2: Periodic Functions and Right Triangle Problems

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Standard Position: The position of an angle with the vertex at the origin and the initial side along the positive horizontal axis.

Coterminal Angles Two angles in standard position are coterminal if and only if their degree measures differ by a multiple of 360°.

Reference Angle: an angle in standard position is the positive, acute angle between the horizontal axis and the terminal side.

φ = θ + (n)360°

Unit Circle: By definition it has a radius of 1

θ θref1 unit

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Background/ExplanationExamplesVocabularyWorking degrees, minutes, seconds

This is all about borrowing just like we do with subtraction in 3rd grade.

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Period Function: A function that repeats itself (y values = the outputs) every so often.

Period: The x values for which it takes the function to repeat itself

Sin Θ = oppositehypotenuse

Cos Θ = adjacenthypotenuse

Sin 600

hypotenuse

adjacent

opposite

Given the unit circle with hypotenuse (of our reference triangles) = 1, The sin is really opposite length which gets bigger initially heading towards 1 at 90o and then smaller heading towards 180o. The sin graph demonstrates the same thing.

SineCosineTracer1 Lesson 2.3.gsp

Link

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hypotenuse

adjacent

oppositeAS

T CAll StudentsTakeCalculus

memory tool for which trig function and positive or negative in each quadrant.

The sign's are really determined by the values of u and v (the same thing as x and y in the coordinate plane)

The ordered pair where the terminal side of ANY ANGLE intersects the unit circle.

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Draw a 147o angle in standard position. Mark the reference angle and find its measure. Then find cos 147o and cos Θref What is the relationship between the two values.

An angle and its ref angle will always have the same magnitude. The quadrant of the angle determines its sign.

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Assume this is not a unit circle for right now, but that the legs are 8 and 5 with an angle terminating in the 4th quadrant. Find the sin and cos

Find by using the pathagorean theorem

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Background/ExplanationExamplesVocabulary Let (u, v) be a point r units from the origin on the terminal side of a rotatingray. If is the angle in standard position to the ray, then the following hold.

Evaluate by calculator the six trigonometric functions of 58.6°. Round to four decimal places.

Each has C & S

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Background/ExplanationExamplesVocabulary Let (u, v) be a point r units from the origin on the terminal side of a rotatingray. If is the angle in standard position to the ray, then the following hold.

The terminal side of angle contains the point (–5, 2). Find exact values of thesix trigonometric functions of θ. Use radicals if necessary, but no decimals.

This is not a Unit Circle

Make + or decision based on the quadrant of the terminal side.

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The triangles (the angles)we need to know exact values for

cosθ2 means cos2 θWe should find the cos of the angle and then square the result.

Do not square the angle and find the cos of that result

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The axis angles:

90o

180o

270o

360o

How do we know what the sides of the triangle are if they don't form a triangle becasue they have collapsed into 2 sides?

Identify the (u , v) coordinates for the place where the terminal side of these angles intersect the Unit Circle.

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ExamplesVocabulary Background/Explanation

2 Scenarios1st: We have angle measures but not all the side lengths.

Solve using the trig ratios

2nd: We have side lengths, but not the angle measures.

Solve using inverse trig functions of our calculator.

Inverse Trig Function:

sin1.85 = xWhat angle has a sin of .85

cos1.56 = xWhat angle has a cos of .56

tan1.42 = xWhat angle has a tan of .42

We use the inverse trig functions to undo the trig functions

Solve... sin x = .85

Solve... sin x = .85

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The inverse trig function will give us values on what is called the Principal Branch of their graphs.

sin: 90 to 90

cos: 0 to 180

tan: 90 to 90

You have the job of measuring the height of a local water tower. Climbing makes you dizzy. You decide to do the whole job from the ground (b/c you learned trig in high school). You find that from a point 47.3 meters from the base of the tower you must look up (angle of elevation) at an angle of 53o to see the top of the tower. How high is the tower?

We choose to work with _______b/c it involves the ______ (which we need) and the _______ which we have.

Which Trig Ratio should we use?

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A ship passing through the Straits of Gibraltar. At its closest point of approach, Gibraltar radar determines that the ship is 2400 meters away. Later, radar determines that the ship is 2,650 meters away.

a) By what angle θ did the ship's bearing from Gibraltar change?

b) How far did the ship travel between the two observations?

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How tall is the flappole if it casts a shadow 11.6 meters with an angle of elevation from the end of the shadow of 36o ?

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Pyramid Problem: The Great Pyramid of Cheops in Egypt has a square base 230 mon each side. The faces of the pyramidmake an angle of 51°50′ with the horizontal(Figure 25p).

a) How tall is the pyramid?

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Pyramid Problem: The Great Pyramid of Cheops in Egypt has a square base 230 mon each side. The faces of the pyramidmake an angle of 51°50′ with the horizontal(Figure 25p).

b)What is the shortest distance you would have to climb to get to the top?

Getting out of Degrees, Minutes, Sec

Enter 45o 32' 14" using the angle menu and alpha + and hit enter and it gives you decimal.

Getting from Decimal to DMS. Enter decimal and then use angle menu DMS.,

3 Strategies from here...

1) 2) 3)

Which is best in this case?

The ______ ratio is our best bet b/c it doesn't involve using an answer WE came up with earlier in the problem.

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Scenario 3:

We will need to set up more than 1 equation with the info we have. How do we solve groups of equations that the use the same variables?

These are called system: Use1) Substitution: Define one of your variables in terms of the other.2) Elimination3) Graphing

How tall is the object?

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Unit 3: Applications of Trig and Circular Functions

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Amplitude: Half the disctance from your min y to your max y

ymax ymin

2Cycle: The part of the graph that is repeating itself again and again....

Period: The total value of x or θ that it takes for a cycle to be completed

Changing Amplitude:

Changing Period:

Phase displacement (or Shift):

Phase Displacement (or Shift): shifting or translating a graph in the x directions.

How does this new vocabulary relate to the transformations we have been dealing with?

Sinusoidal Axis: The centerline of the graph. The sinsusoidal axis moves from y = 0 based on vertical translation. It will be found at y = halfway between max and min

Sinusoidal Axis:

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General Form of Sinusoidal Functions (Algebraically)

y = A cos B(x D) + C

y = A sin B(x D) + C

What do the letters A, B, C, and D do or effect?

A is the amplitude

B effects the period, its reciprocal is the factor by which we x dilate.

C is our y translation and it creates our sinsusoidal axis.

D: Horizontal Translation which we are also calling phase displacement of phase shift.

Period/ Frequency:

Period:

Frequency:

Concave / Convex

Point of Inflection: a point at which our graph changes from concave to convex or vice versaAlways on the sinusoidal axis.

D/B if the B has not been factored outside of the ( ).

Reciprocals

Identify the parts

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Period/ Frequency:

Period:

Frequency:

Concave / Convex


Suppose that a sinusoid has a period of 12° per cycle, an amplitude of 7 units, a phase displacement of –4° with respect to the parent cosine function, and asinusoidal axis 5 units below the xaxis. Without using your grapher, sketch this sinusoid and then find an equation for it. Verify with your grapher that yourequation and the sinusoid you sketched agree with each other.

Cosine equations are easier to work with. Why?

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ExamplesVocabulary Background/ExplanationCosine equations are easier to work with.

• What did it take to move the point (0,1), a critical point on the cosine function, to our initial maximum?

Finding the horizontal translation change when converting from cos to sin.Prior to a change in period the translation from cos to sin is right 90o, or 1/4 of the period.

As our Period gets dilated, so must our 90o translation.Formula

To work with sine equations.• What did it take to move the point (0,0) to the closest point of inflection (because this point is on the sinusoidal axis where sine functions start)

Or

Cos to Sin: +(1/4)(Period)

Sin to Cos: (1/4)(Period)

Period/ Frequency:

Period:

Frequency:

Concave / Convex


Write a particular equation for the following and identify the parts.

Amplitude:

Phase Displacement

Sinusdoidal Axis

Period

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ExamplesVocabulary

Period/ Frequency:

Period:

Frequency:

Concave / Convex


Write a particular equation for the following and identify the parts. This is a quarter cycle.

By knowing that it is a quarter cycle, we can sketch the remaining part of the cycle.

Amplitude:

Phase Displacement

Sinusdoidal Axis

Period

Write another equation with a phase dispacement other that 10o

As cos

As Sin

Background/ExplanationCosine equations are easier to work with.


Finding the horizontal translation change when converting from cos to sin.Prior to a change in period the translation from cos to sin is right 90o, or 1/4 of the period.

As our Period gets dilated, so must our 90o translation.Formula


Or

Cos to Sin: +(1/4)(Period)

Sin to Cos: (1/4)(Period)

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Period/ Frequency:

Period:

Frequency:

Concave / Convex


Cosine equations are easier to work with.


Write a particular equation for the following and identify the parts. This is a quarter cycle.

By knowing that it is a quarter cycle, we can sketch the remaining part of the cycle.

Amplitude:

Phase Displacement

Sinusdoidal Axis

Period


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Period/ Frequency:

Period:

Frequency:

Concave / Convex


Cosine equations are easier to work with.


Amplitude:

Phase Displacement

Sinusdoidal Axis

Period To work with sine equations.• What did it take to move the point (0,0) to the closest point of inflection (because this point is on the sinusoidal axis where sine functions start)

Graph the following..

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Background/ExplanationExamples

Lesson: 3.3 Examples: 1 2 +

Objective: SWBAT plot the graphs of all trig functions.

Sin and Csc

Cos and Sec

You can graph every single csc graph by first graphing or imagining the related sin graph.

Csc = 1 / sin so everytime our related sin function is 0, we have an asymptote.

Our Peaks meet our valleys.

You can graph every single sec graph by first graphing or imagining the related cos graph.

Sec = 1 / cos so everytime our related cos function is 0, we have an asymptote.

Our Peaks meet our valleys.

45 180 360

12

12

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Tan as...

Cot as ...

sincos

cossin

The period of the tangent function is 1800. If you always make your x scale into 1/4's of your period, you can graph tangents easier because you have 3 points. The "origin" and over 1 gridline, up 1 value.

The period of the cot function is 180o. Again, make scale of the x axis 1/4 of the period for easy graphing.

Asymptotes occur where ...

Asymptotes occur where ...

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Graph y = 3csc 3(x 45) 4

If we graph y = 3 sin 3(x 45) 4 our peaks will meet our valley.

To graph by hand you need to first draw or imaging the sin graph. Then your peaks just meet your valleys.

To graph on calculator you need to be VERY CAREFUL and only take the reciprocal of the trig calculation so vertical translation and change in amplitude are put out front.

y = 3sec4(x +10o )+1a) Identify the aspects of the related sinusoidal function

Consider the cosine: amplitude = sinusloidal axis y = Period = Phase Shift:

b) Graph by handc) Confirm with calculator

30 120 240 300

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Why are there 2π radians in a unit circle?

As we wrap a number line around the unit circle, we would find that the circumference of the circle is 2π b/c the circumference is 2πr and the radius is 1. 2π is approx 6.28

So a radian is really a measurement of the arc length / angle creating an arc length.

Radian = arc length radius

Converting from Degrees to Radians...

Degree X 180oπ

So we are really asking ourselves what fraction of 180o is our degree measurement.

45o

Converting from Radians to Degrees...

Radian (with π ) X180o

π So we are really just asking ourselves what is the fraction part of 180o

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What degrees is 5.73 radians?

5.73 radians

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Find the radian measure and degree measure of the angle whose sine is 0.3

Think of the equation that this implies.

Solve using inverse trig functions.

If there is no degree symbol than it is ASSUMED to be a measurement in radians. Make sure you are writing what you mean!!!

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We need to know the quick conversion for the following...

Graph...

in order to graph this we should consider the following:

Amplitude:

Sinusoidal Axis :

Period :

Phase Shift:

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It is NOT TRUE that

csc 1 = 1 / sin1 or

sec1 = 1/ cos 1 or

cot1 = 1/ tan1

Instead, we need to draw a right triangle and find a way to use the sin1, cos1, or tan1

Find the csc1 1.001

What angle has a csc of 1.001?

csc X = 1.001

csc = hypotenuse/ opposite so we need to make our ratio of h / o have a value of 1.001

But our calculator doesn't have a csc1 feature, so what should we do?

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Radian = arc length radius

Circular Functions...Circular Function: periodic functions whose independent variable is a real number without any units. Circular Functions are more appropriate for realworld problems.

As we wrap the number line around the unit circle, we can see how a real number can be represented on the independent, x, axis.

Example 1: Plot the circular function on your grapher. Determine the period both graphically and algebraically.

y = 4 cos 5x

Graphically, we can use the Calc Max feature to determine the first x value at the end of our period.

Algebraically, we need to use....

Period = Normal Period of Parent Function B

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ExamplesVocabulary Background/ExplanationFind a particular equation for the sinusoid function in Figure 35e. Notice x on the horizontal axis, not θ, indicating the angle is measured in radians. Confirmyour answer by plotting the equation on your grapher.

The book will use x to indicate radians and θ to indicate degrees.

Amplitude:

Phase Displacement

Period:

Sinusoidal Axis

Function:

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Sketch the graph of y = tan (π/6)x.

Identify the aspects....

Horizontal translation

Vertical translation

Y dilation

X dilationNormal Period B = New Period

By imagining the period as being broken into chunks of 4, you can know where to plot the points (x, 1) and (x, 1). If there had been y dilation these values (1 and 1) would be stretched.

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Because there are an infinite number of x values associated with many y values (in this case y = 0.3), we need to come up with a way to define all the answers to the question...

0.3 = cos X

Cos1 is used to find 1 value on the primary branch of the function.

Arccos (arcsin, arctan) will be used to identify ALL the answers.

We will need to use the cos1 feature and our knowledge of ASTC, reference angles, and periods to answer this type of question.

Find the arccos 0.3.

The general solution for the arccosine of a number is written this way:

Formula for Arccos I don't recommend learning this concept on formula

Arccos 0.3 means any angle whose cos is 0.3

v= cos1 0.3

= 1.2661

v= cos1 0.3

= 1.2661

u = 0.3

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0.4 = cos X

Find graphically the six values of x for which y = 5 for the sinusoid above.

Estimate the values by drawing vertical lines down to the xaxis.





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0.4 = cos X


Find it numerically, (Table of Values)

Amplitude =

Sinusoidal Axis :

Period:

Phase Displacement:

Identify Parts in order to write function....

Function:

Put function in calculator and look in table for when y = 5




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0.4 = cos XFind it algebraically (Table of Values)

Function:





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0.4 = cos XFind it algebraically (Table of Values)

Function:

Our point on the rise where y = 4 must be equal distance from the axis of symetry x = 10.5 as is our point on the decline where y = 4. That distance is 10.58.51 = 1.99. So 10.5 + 1.99 = 12.49





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Waterwheel Problem: Suppose that the waterwheel rotates at 6 revolutions per minute (rpm). Two seconds after you start a stopwatch,point P on the rim of the wheel is at its greatest height, d = 13 ft, above thesurface of the water. The center of the waterwheel is 6 ft above the surface.

a. Sketch the graph of d as a function of t, in seconds, since you started the stopwatch.

b. Assuming that d is a sinusoidal function of t, write a particular equation.Confirm by graphing that your equation gives the graph you sketched inpart a.

c. How high above or below the water’s surface will P be at time t = 17.5 sec? At that time, will it be going up or down?

d = __________ going up. Put function in calculator and look in table for x = 17.5

Example Develop the Function

d. At what time t was point P first emerging from the water?We could plot the function in y1 and calc the zero we are looking for. Or we could use Algebra...

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SWBAT

Unit 4: Trig Function Properties and Parametric Functions

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y = cos2 x

y = sin2 x

How does this help us see the pythagorean property which tells us that ...

sin2x + cos2x = ____

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Properties of trig functions are going to be useful for us to eventually be solving trig equations. Also, they may allow us to calculate values that we didn't think we had.

Reciprocal Properties:

Restrictions: Denominators can't be 0

Identity: An equation that is true for every value of he variable/s

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Quotient Properties

Quotient means the result of division

Restrictions: Denominators can't be 0

Write tan x in terms of sec x and csc xIdentity: An equation that is true for every value of he variable/s

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Pythagorean Properties

Based on right triangles for which we know that u2 + v2 = r2


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1) Reciprocal Properties:

2) Quotient Properties

3) Pythagorean Properties


In order to transform/simplify trig expressions we can use the following properties...

as well as factoring and other math operations.

Transform

into

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Transform

into

5








Transform

into

3


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ExamplesVocabulary Background/ExplanationSolve

10 sin (x 0.2) = 3 for the domain [0,4π]

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The relationship between the arc function and the inverse function ...

Solve

θ = arctan(4)θ∈[0o, 720o]

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Solve the equation 4tan 2θ = 5 for the first three positive values of x. Verify solutions graphically.

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ExamplesVocabulary Background/ExplanationQuadratic Forms

Solve...

cos2 θ + sin θ + 1 = 0for θε[-90o, 270o )

When solving a trig or circular equation that initially has more than 1 function in it, try to rewrite so that it is all in terms of 1 function.

Replaced cos2 θ with its equivalent.

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If a variable appears both transcendentally (in the argument of the function) and algebraically (not in the argument) then our trig equation likely CAN NOT be solved algrebraically.

This is b/c we are not able to transform the equation into a single function....

f(argument) = constant

for all real values of x

In these cases we set...

y1 = left sidey2 = right sidc

and we look for the points of intersection.

Solve by Graphing

9


10

Solve

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If two related variables x and y both depend on a third, independent variable t,the pair of equations in x and t and y and t is called a parametric function.

Parameter: The independent variable t in a parametic equation

The position of the pendulum could be described by...

Example 1: Plot the parametric function in degree mode...

x = 5 cos ty = 7 sin t

This is an ellipse.

The parameter T is NOT AN ANGLE IN STANDARD POSITION. Rather, t is generally time and the x and y positions are functions of time.

what shape is this?

4

ExamplesVocabulary Background/ExplanationExample 2

Sometimes we can discover properties of a graph by eliminating the parameter, creating a single Cartesian equation with only x and y.

For the parametric function x = 5cos t and y = 7sin t eliminate the paramenter to get a Cartesian equation relating x and y. Describe the graph

Remeber that x is related to cos and y related to sin based on the triangle in a unit circle.

To eliminate the parameter use the fact that cos2t + sin2t = 1 so let's square both sides of the given equations and then add.

Cartesian Forms

Circle: x2 + y2 = C

Ellipse: ax2 + by2 = C a≠b

Hyperbola: ±x2 ±y2 = C x and y have different signs

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ExamplesVocabulary Background/ExplanationDescribe the effects of the constants 6 and 3 on the graph. Parametric Functions....

x = h + a cos t

y = k + b sin t

h and k are causing translation.

h causes left/right + h moves right h moves left

k causes up/down + k moves up k moves down

So... (h, k) becomes the center of the conic section

a and b are the radii of the conic section. If a = b then you have a circle.

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Parametric Functions....x = h + a cos t

y = k + b sin t

h and k are causing translation.

h causes left/right + h moves right h moves left

k causes up/down + k moves up k moves down

So... (h, k) becomes the center of the conic section

a and b are the radii of the conic section. If a = b then you have a circle.

Figure 45d shows the outlines of a cylinder. Duplicate this figure on your grapher by finding parametric equations of ellipses to represent the bases and then drawing lines to represent the walls.

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Design the following on your grapher and write down the parametric equations used.

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Design the following on your grapher and write down the parametric equations used.

Ellipse (all solid)

x = 13+ 0.8cos Ty = 6 + 4sin TStart the Draw Line feature at (3,6)

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Tool for remembering the principal branches of the inverse trig functions.

This is a memory tool for remembering the principle branches of the inverse trig functions.

x y x y x y

is equivalent to...

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Using Parametric Mode to graph inverse trig functions.

x = sin (T)y = T

b/c this tells us that y is T and we are doing x = sin y which is the inverse (switch x and y) of y = sin x.

A T min and max of Pi/2 would create only the principle branch

5




6




7


Evaluate the

use radians

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SWBAT

Unit 5: Properties of Combined Sinusoids

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y = 4sin x

y = 3cos x

y = 4sin x + 3 cos x

Is the sum of sinusoids also a sinusoid?

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The graph of ..

y = 3 cos θ + 4 sin θ

is a sinusoid that can also be written as ...

y = A cos (θ D)where...

A = amplitudeD = phase displacement (x value of "1st" maximum compared to x value of 0)

D can also be expressed as an angle in standard position with sides of u = 3 (coefficient of cos) and v = 4 (coefficient of sin) 4

3

A = 5D = 53.13

A is the hypotenuse of the reference triangle of D.

Express y = 8 cos θ + 3 sin θas a single cosine with a phase displacement

We need to find A and D...

for when n = 1 b/c this puts the angle in the correct quadrant which we already know to be quadrant II The quadrant of D is detemined by c

and b.

=

4


Express 7 cos (θ - 23o ) as a linear combination of cos θ and sin θ

In lesson 5.1 we discovered that

cos (θ - D)≠cos θ - cos Dcos (θ + D)≠cos θ + cos D

so the distributive property doesn't apply to cos.

We can still express...

cos(θ - D) in terms of cosines and sines of θ and D

Composite Argument Property for cos (A - B)

cos (A - B) = cosAcosB + sinA sinB

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ExamplesVocabulary Background/ExplanationSolve 5 cos θ + 7 sin θ = 3 for x in the domain [0, 2π]

We have used the technique of converting from a linear composition of functions to a single cos function in order to create and equation with 1 function.

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ExamplesVocabulary Background/ExplanationSolve 2 cos x + 3 sin x = 2 for x in the domain [2π, 2π]

We have used the technique of converting from a linear composition of functions to a single cos function in order to create and equation with 1 function.

3

ExamplesVocabulary Background/ExplanationOdd/Even Properties:

Even Functionscos (x) = cos (x) sec (x) = sec (x)

Reflected over the ____________

Odd Functionssin(x) = sin(x) csc(x) = csc(x)tan(x) = tan (x) cot (x) = cot(x)


Cofunction Propertiescan also be expressed in radians

cos x = sin (90o x) sin x = cos (900 x)cot x = tan (90o x) tan x = cot (90o x)csc x = sec (90o x) sec x = csc (90o x)

Composite Argument Properties

cos (A B) = cosAcosB + sinAsinBcos (A +B) = cosAcosB sinAsinBsin (A B) = sinAcosB cosAsinBsin(A + B) = sinAcosB + cosAsinBtan (A B) = tan A tan B 1 + tanAtanBtan (A + B) = tan A + tan B 1 tanAtanB

Solve the equation for x in the domain x ε [0,2π] and verify the solution graphically.

sin 5x cos 3x cos 5x sin 3x = 1/2Strategies for solving trig equations

1) What property does it look like is being used? Perhaps rewriting the equation using that property will be helpful b/c we already know how to solve trig equations that have only ONE trig function in them.

4






use the composite argument property to show that the given equation is an identity

Odd/Even Properties:





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ExamplesVocabulary Background/ExplanationOdd/Even Properties:







Solvecos 3x cos x + sin 3x sin x = 1 for x [0,2π]

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ExamplesVocabulary Background/ExplanationShow by example that these are equivalent.

cos θ = sin (90o θ)

3

ExamplesVocabulary Background/ExplanationSum and Product Properties—Product to Sum

2 cos A cos B = cos (A + B) + cos (A – B) –2 sin A sin B = cos (A + B) – cos (A – B) 2 sin A cos B = sin (A + B) + sin (A – B) 2 cos A sin B = sin (A + B) – sin (A – B)

A system of equations: 2 or more equations using the same variables. Can be solved using...

substitution elimination graphing matrices

Transform 2 sin 13° cos 48° to a sum (or difference) of functions with positivearguments. Demonstrate numerically that the answer is correct.

Which property to the right applies?

Sum and Product PropertiesSum to Product

sin x + sin y = 2 sin0.5(x + y) cos0.5(x – y) sin x – sin y = 2 cos0.5(x + y) sin0.5(x – y) cos x + cos y = 2 cos0.5(x + y) cos0.5(x – y) cos x – cos y = –2 sin0.5(x + y) sin0.5(x – y)

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Transform cos7θ cos 3θ to a product of functions with positive arguments.



5





Transform

2sin8xcos2x into a sum or difference of sines or cosines with positive arguments



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Solve

Sin 3x Sin x = 0



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Solve

sin 3θ + sin θ = 0 for θ [0o, 360o]



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Provecos(x+y)cos(xy) =cos2 x sin2 y



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ExamplesVocabularyPROPERTIES: Products and Squares of Cosine and Sine

Double Argument Properties sin 2A = 2 sin A cos A

cos 2A = cos2 A – sin2Acos 2A = 2 cos2A – 1cos 2A = 1 – 2 sin2A

Half Argument Properties

Background/Explanation

If cos x = 0.3, find the exact value of cos 2x. Check your answer by finding the value of x, doubling it, and finding the cosine of the resulting argument.

b/c we have cos xx = arccos 0.3 + 2πn

Old Way New Way

cos 2x = 2 cos2x 1

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ExamplesVocabulary Background/ExplanationWrite an equation expressing cos 10x in terms of sin 5x PROPERTIES:

Products and Squares of Cosine and Sine




5


Write and equation expressing cos 10x in terms of sin 5x using the composite argument property for cosine.

PROPERTIES: Products and Squares of Cosine and Sine




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ExamplesVocabulary Background/ExplanationIf cos A = 15/17 and A is in the open interval (270o, 360o )

a) Find the exact value of cos 1/2 A

b) Find the exact value of cos 2 A

c) Verify numerically.

PROPERTIES: Products and Squares of Cosine and Sine




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SWBAT

Unit 6 Triangle Trigonometry

3


The Law of Cosines

The law of cosines can be used to determine missing sides and angles. Using it to find an angle we need to be particularly careful with our calculations.

In triangle PMF, angle M = 127°, side p = 15.78 ft, and side f = 8.54 ft. Find the third side, m.

4


The Law of Cosines


In triangle XYZ, side x = 3 m, side y = 7 m, and side z = 9 m. Find the measure of the largest angle.

5


The Law of Cosines


Suppose that the measures of the sides in Example 2 had been x = 3 m, y = 7 m,and z = 11 m. What is the measure of angle Z in this case?

3


Area of a Triangle....

Old School: Area = (1/2) b h

You need to have 2 sides and the included angle.

Find the Area of the triangle

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Area of a Triangle....

Old School: Area = (1/2) b h

You need to have 2 sides and the included angle.

Find the area of Triangle JDH if j = 5 cm, d = 7 cm, and h = 11 cm.

What am I missing?

5


Hero's Formula allows you to find the area of a triangle directly from the length of the 3 sides. No included angle needed as with our Area Formula.

Semiperimeter: half the perimeter

Find the area of Triangle JDH if j = 5 cm, d = 7 cm, and h = 11 cm.

Can't I find the area without first finding an included angle?

6

Examples Background/Explanation

3


Law of Sines

Normal Case:

When the angle is enclosed by the 2 sides, you don't have to worry about ambiguity.

Law of Sines vs. Law of CosinesThe law of sines allows you to work when you only know 1 angle and 2 sides.

Ambiguous Cases:

Given 2 sides and a nonenclosed angle.

You have a triangle with sides 80 cm and 50 cm and a nonenclosed angle of 26 degrees. Draw the triangle.

Find the remaining side

This creates a quadratic equation ...

use quadratic formula, or graph...

We must use the law of cosines in this situation and set up a quadratic equation to solve. There will be 2 answer as long as the discriminant (b2 4AC) is positive. There will be no triangles if the discriminant is negative.

Which triangle are we talking about?

Law of Sines Law of Cosines as Quadratic

4


Law of Sines

Normal Case:

When the angle is enclosed by the 2 sides, you don't have to worry about ambiquity.

Law of Sines vs. Law of CosinesThe law of sines allows you to work when you only know 1 angle and 2 sides.

Ambiquous Cases:

Given 2 sides and a nonenclosed angle.

You have a triangle with sides 90 cm and 30 cm and a noninclosed angle of 26 degrees. Draw the triangle.

Because there are two possible triangles the result from this description, we need to be careful with the law of sines in this scenario.Find the remaining parts

3


Making our own decisions as to what to use for a story problem involving triangles.

Law of Cosines• Find the third side from two sides and the INCLUDED angle.• Find angle if you have 3 sidesFind both values of the missing side in the ambiguous case.

Law of Sines• ASA or AAS• You can (not recommended) use it to find an angle, but there are 2 values of the arcsine between 0 and 180 that could be the answer.• You can't use it for SSS case.• You can't use it for the SAS case b/c the opposite angle is unknown.

Area Formula• Need 2 sides and included angle

Hero's Formula• Find area from 3 side lengths.

What are the lengths of the other 2 parts of the roof?

Studio Problem: A contractor plans to build an artist’s studio with a roof that slopesdifferently on the two sides (Figure 67a). Onone side, the roof makes an angle of 33° withthe horizontal. On the other side, which has awindow, the roof makes an angle of 65° withthe horizontal. The walls of the studio areplanned to be 22 ft apart.

How many sf of paint will be needed for each triangular end of the roof?

4


Making our own decisions as to what to use for a story problem involving triangles.

Law of Cosines• Find the third side from two sides and the INCLUDED angle.• Find angle if you have 3 sidesFind both values of the missing side in the ambiguous case.

Law of Sines• ASA or AAS• You can (not recommended) use it to fin an angle, but there are 2 values of the arcsine between 0 and 180 that could be the answer.• You can't use it for SSS case.• You can't use it for the SAS case b/c the opposite angle is unknown.

Area Formula• Need 2 sides and included angle

Hero's Formula• Find area from 3 side lengths.

Detour Problem: Suppose that you are the pilot of an airliner. You find it necessary todetour around a group of thundershowers, asshown in Figure 67b. You turn your plane atan angle of 21° to your original path, fly for awhile, turn, and then rejoin your original pathat an angle of 35°, 70 km from where you left it.

How much farther did you have to go because of the detour?

We traveled _______ km further.

What is the area of region enclosed in the triangle?

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Examples Background/Explanation

Rocket Problem: An observer 2 km from the launching pad observes a rocket ascending vertically. At one instant, the angle of elevation is 21°. Five seconds later, the angle has increased to 35°.

How far did the rocket travel during the 5 second interval?

Assuming constant average spped, what is the angle of elevation 15 seconds after the first sighting?

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ExamplesVocabulary Background/ExplanationUnderwater Research Lab Problem: A ship is sailing on a path that will take it directly over an occupied research lab on the ocean floor. Initially, the lab is 1000 yards from the ship on a line that makes an angle of 6° with the surface (Figure 67c). When the ship’s slant distance has decreased to 400 yards, it can contact people in the lab by underwater telephone. Find the two distances from the ship’s starting point at which it is at a slantdistance of 400 yards from the lab.

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SWBAT

Unit 7: Properties of Elementary Functions

3

ExamplesVocabulary Background/ExplanationConsider the graph...

a) What kind of function is this?

b) On what interval or intervals is the function increasing or decreasing? Which way is the graph concave?

The function refers to the y value, what is happening to it as x increases.

c) Describe something in the real world that a function like this could model.

d) Find the particular equation....Strategies:

1) Use a linear form (standard, pointslope, slopeintercept) and substitution2) Set up a system and solve with matrix.3) Regression

4


a) Type of the function?

b) Over what intevals is the function increasing/decreasing and describe the concavity.

c) Real World?

d) Develop the particular function. Strategies....1) System of equations and solve with a matrix.2) Regression.

5


a) Type of function?

When doing the regression, don't put (0,0) in the lists as this will mess up the PwrReg, just use nonorigin known points.


c) Develop the particular function. Strategies....1) System of equations2) Regression.

6


a) Type of function?


c) Develop the particular function. Strategies....1) System of equations2) Regression.

3

ExamplesVocabulary Background/ExplanationPattern???

Add Add Pattern describes a _______________. Occurs when every time some value is added to the x coordinate another constant is being added to the y coordinate.

4


Pattern???

AddMultiply Pattern describes an . Every time

a certain amount is added to the x coordinate, the y coordinate is multiplied by a constant.

5


Pattern???

MultiplyMultiply Pattern describes a _______________, but it can be helpful to ignore a data point in order to see the multiplication pattern that is happening in the x coordinates.

If we developed the particular function, the point we ignored would still be on the graph, we only "ignored" it in order to see the pattern of multiplication in the x's.

6


Pattern???Second Difference Pattern represents a ________________. The pattern has addition in the x's and if we do subtraction in the y's the next comparison will yield a constant number.

7


Pattern???

a) Decide which type of function this is by identifying a pattern.

b) Develop the particular function.

8


f(5) = 12f(10) = 18find the f(20) if f is...a) Linear

b) Power

c) Exponential

9


Describe the effect on y of doubling x if ...

a) y varies directly with xDirect Variation:

y = kx the key being varying directly implies multiplication.

b) y varies inversely with the square of x.Inverse Variation:

y = k/x. Varying inversely implies division.

c) y varies directly with the cube of x.

3


Logarithms are inverses of exponentials.

Converting Between the Two

x in this log is known as the argument.

Common Logarithm:

Log with base 10

when no base is written, base 10 is implied.

Natural Logarithm:

Log with base e and is written ln

4


3 Properties of Logarithms:

1) Power Property of Logarithm

Allows us to get variables out of the exponent position.

2) Product Property of Logarithms

3) Quotient Property of Logarithms

Demonstrate that log 3 + log 5 = log ( 3 * 5)

Simplify (make into a single log)

5


Solve the equation...

We solve log equations by making them into a single log of a single argument and then converting them to exponential expressions.

Conversion Process

6


Solve the equation...

4 3 ln (x + 5) = 1 We solve log equations by making them into a single log of a single argument and then converting them to exponential expressions.

Conversion Process

13


Change of Base Formula:

Base and Argument Property:

Argument of 1...

14

ExamplesVocabulary Background/ExplanationSolving Exponential Equations

4 Strategies:

1) Graphing2) Taking Logs and using the power property.3) Common Base4) Recognize as quadratic.

Solve 73x = 983

Taking Logs

Helps get the variable out of the exponent position by using the power property.

15


4 Strategies:


Solve 3 x 2 = 9 x + 1

Common BaseIf my bases are the same and we are =, then my exponents must have the same value.

16


4 Strategies:


Recognize as quadraticSolve e2x 3ex + 2 = 0

17

Untitled

Grade:«grade» Subject:

Date:«date»

18

1 .

A B C D

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2 .

3


Forms of Logarithmic Equations

The MultiplyAdd pattern fits a logarithmic relationship. The x's are being multiplied and some constant is being added to the y's.

Untranslated Form:

y = a + b log c x

Translated Form:

y = a + b log c (x d)

Develop the particular equation for this function

We multiplied our x's by 3, and it produced a 2 for y looking at the first couple points

x y

6 1

8 2

54 3

162 4

4



Untranslated Form:

y = a + b log c x

Translated Form:


5



Untranslated Form:

y = a + b log c x

Translated Form:


6

ExamplesVocabulary Background/ExplanationPlot the graphs of these functions and identify their domains.

a) f(x) = 3 log (x 1)

The domain of the parent logrithmic function is x > 0.

If we recognize translation, etc we can really identify the domain of a log function without graphing.

b) f(x) = ln (x + 3)

c) f(x) = log 2 (x2 1) The argument of a log can't be negative so we can write an inequality and solve it for the domain.

Exponential

Logarithmic

swbat - st. joseph high school

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