Precalculus 30

Welcome to the Precalculus 30 page.  Here you’ll find links to videos, notes and assignments.

Review Section – This will list a bunch of links (ok, just one for now) to help you review basic concepts.

Six basic functions  – Takes you through the graphs of f(x)=x, x2, x3, square root of x, cube root of x, and absolute value of x.

Unit 1 – Transformations and Functions

Chapter 1 – Function Transformations

1.1 Vertical and Horizontal Transformations of Functions.  Shorter explanation, moves basic graphs around.  (Good)  More detailed explanation, shows tables of values as well as graphs.  (Better)

1.2 Reflections and Stretches  Shows how to compress or stretch the graph of f(x) horizontally and vertically.  Watch this one first!  Shows how to reflect a graph about the x-axis and about the y-axis.

1.3  Combining Multiple Transformations  Video 1 of 2.  Video 2 of 2.  This is a summary of what a, b, c (or h), and d (or k) do for a function, f(x), as it is translated, reflected, compressed and stretched, using the formula af(b(x-c))+d

1.4 Inverse of a Relation Little wordy, but does a thorough job of describing the inverse of a function.  Video 1 of 2.  Part 2 of 2…

2.1 Radical Functions

The important thing to remember here is that the radical function is set up as follows:  f(x) = a√b(x-c) +d.   (Note:  The square root goes over the b(x-c) – limitations of the blog editor rear their ugly head…)  a, b, c, and d still do the same things.

a – vertical compressions (if it’s between 0 and 1); vertical stretches (if it’s greater than 1) and reflection about the x-axis (if it’s negative).

b – horizontal compressions (if it’s greater than 1 – note the difference from how this works with a, above); vertical stretches (if it’s between 0 and 1) and reflection about the y-axis (if it’s negative)

c – moves the function left or right. (note: if it’s x-c, then it moves to the right c units {this corresponds to c being positive – and yes, this is confusing because it’s x – c, so x is subtracting a positive.  So, if it’s x + c, then it moves to the left: it is x + c because you are subtracting a negative, so c is negative and it moves left – clear as mud, right?  Bottom line : x – c moves right, x+c moves left.)

d – moves the function up or down ( + d moves it up d units, – d moves it down d units)

We’ll do some examples in class…

Homework / Pg 75 / 3, 5, 9, 10, C1

2.2 – Square Root of a Function. – decent video for finding the domain of a square root function.

Notes for the rest of the function :Section 2-2 Notes pdf

Homework / Pg 86 / 3-8, 10-12

2.3 Solving Radical Equations Useful video.  He does 2 different examples solving equations with square roots in them.

Section 2-3 Notes   Notes for your benefit.

Homework / Pg 96 / 1, 3, 5-8, 12, 16, 17

3.1 Characteristics of Polynomial Functions

Couple of videos to watch.  Summary, before you start watching.

First, a ROOT of a function is a place where it crosses the x-axis.  Second, often these videos will use INTERVAL NOTATION.  This means that, instead of writing -3 < x < 4, you would just write (-3,4).  It looks kind of like they’re naming a point, except that they’re using this notation to give a range of x-values.  If they use a ≤ or ≥ sign, then they’d use a square [] bracket instead.  So, -3 < x ≤ 4 would be (-3,4].  If you are listing 2 different ranges of x-values, (Ex.) x<-3 and x≥8, then we’d write it like this:  (-∞,-3)U[8,∞). 

Since infinity is a concept, and not an actual number, you can’t actually get there, so it will ALWAYS have a round bracket () at it.  Good?  Good.  Good video – only watch first 7:30 of it, ignore last 2 minutes.  Also a good video – gives basic shapes, shows how to match different shapes to their equations.  Only watch the first 7:10.  Ignore the last couple minutes again…

Homework / Pg 114 / 1-4 all, 7, 12 a) and b)

3.2 – The Remainder Theorem – Dividing Polynomials by Binomials    Good video on how to divide various polynomials by binomials using long division.  Synthetic Division examples.  I’m not a big fan of synthetic division (personal bias), but somepeople think it’s fan-foo-goo-tastic, so give it a try.

Homework / Pg 124 / 3-8 all, 14, 15, C3

Section 3.3  The Factor Theorem.

Section 3-3  Notes for your viewing pleasure.

Homework / Pg 133 / 2-7, 16 a), C1

3.4  Sketching Polynomial Graphs  Good video.  Does an x^5 polynomial.  Spends the first 9 minutes factoring it, so if you know how to do that, skip ahead.


A Note on End Behaviour – to figure out how a function behaves at the ends (really big positive, or really big negative), all you have to look at is the degree (is it a parabola, cubic, quartic, etc) AND the value of the leading coefficient (the number that you multiply the highest order term by).  So, for instance, if you have -3x^4 + blah blah blah, you look at the x^4 – quartic.  Multiply that by -3, and you know that your graph will point DOWN, because it will be flipped about the x-axis.

We’ll talk about root multiplicity in class.

4.1 and 4.2  We did this in class with the “hoopstick” activity.

Homework – Pg 175 / 2-8, 12, 13  ALSO  Pg 186 / 2-6 odd letters, 9, 11, C1, C4


4.3 Trigonometric Ratios.  Good video which shows how to sketch an angle on the unit circle and find sin x and cos x.

Secant, Cosecant and Cotangent – attempt at an explanation…Reciprocal Trigonometric Functions

Homework – Pg 201 / 1, 2, 5, 6, 8, 10, 12, 18

4.4 Intro to Trig Equations  Good video, but you can ignore the graphs right now, b/c we haven’t done that yet.  This one is just linear equations.  This one does squared trig equations.  Yahoo!

Homework – Pg 211 / 3-7, 9-12, 15, 18, 23

5.1 Graphing Sine and Cosine  Good video dealing with basic graphs of sine and cosine.  At one point when he’s graphing sine, he incorrectly labels 3π/2 as 3π/4, but then he gets it right for cosine.  This video shows what happens when you change the amplitude and period of a sin or cos function by altering A and B in f(x)=Asin(Bx)

5.2 Horizontal and Vertical Translations of Sine and Cosine  Shows, well, basicaly horizontal and vertical translations…?  Yeah…  So, f(x)=sin(x-C)+D, just like we did in Unit 1!

 5.3 Graphing Tangent  The first 2:10 is kind of a history lesson, you can watch it for interest sake if you want, but I don’t require you to know it.  One major difference b/w Tangent and sine/cosine is that Tan has a period length of π, whereas sin/cos has a period of 2π.

We’ll go over graphing a tan graph with A, B, C and D in class.

5.4  Solving Trip Equations Graphically

Section 5.4 Solving Trig Equations Graphically  A little bit of notes from Old Man Herman.  And no, you can’t call me that in class. 🙂

6.1  8 Basic Identities

The Eight Fundamental Identities  PDF file which shows where the 8 basic identites come from – The Reciprocal, Quotient and Pythagorean Identities.

6.2 Sum, Difference and Double Angle Identities

Yes, I know – FOUR VIDEOS?!!?!?  No, I haven’t lost it.  You really only need to watch the first one and the last one.  A summary of all these identities is in your textbook on page 305.  This shows how to use the sum and difference identites for Cosine  This one show how to use the sum and difference identites for Sine.  It’s almost exactly the same as for Cosine, you just use a different identity.  Same deal as the above 2 videos, except for Tan this time.  Really, it’s exactly the same as the first two, you’re just plugging into a different identity.  And finally, this one shows the double angle identities.  Basically, this is just using the sum identity for sin/cos/tan, where you’re adding 2 angles, except the angles are the same – hence the “double angle”.

6.3 Proving Identities

You almost have to think of this like a game.  And you will definately find yourself wondering “when will I ever use this?”  Trust me, when you get into doing integrals with trig functions in Calculus, it’s pretty helpful to be able to substitute easier functions.  You definately won’t use this shopping for groceries or applying for a mortgage, though.  Well, ok, probably not.  Good video showing how to do proofs with the 8 basic identities. As above but using sum/difference / double angle identities.

6.4 – Solving Trig Equations using Identities  Not our usual guy.  Basically, this is very similar to the solving trig equations that we did in 4.4 and 5.4, but this time you need to use identites to make some substitutions before you can solve the equation. – Here’s our regular guy – I was having trouble getting this one to load, though, but then it started working.  You takes your chances, kidlets….  You can ignore the last example in this video where he’s using the quadratic formula, though.

 7.1Characteristics of Exponential Functions  Great video showing exponential functions.  Don’t worry about concavity – we’ll get to that in Calculus next term.

7.2 Transformations of Exponential Functions  Video that shows the effects of C and D, from f(x) = AeB(x-C)+D.  This one show all the values, A, B, C and D HOWEVER, they sometimes mix up what they’re calling them, so just look at the position of the value and compare it to f(x) = AeB(x-C)+D to make sure you know what they’re talking about.  It’s all pretty clear when they put in actual numbers.

7.3 Solving Exponential Equations  You can use this method as long as the bases are the same OR can be written so that they’re the same.  If the bases aren’t the same, then you basically need to graph it / and or do trial and error.  Will do an example of this in class.

 8.1 Understanding Logarithms  Totally cheesy video that simply explains how logs are just powers in disguise.  Me luvit!  Our old favorite guy shows us how the graphs of exponential functions and graphs of logarithmic functions relate.

We’ll go over estimating the values of logs in class.

8.2 Transformations of Log Functions.

Just like with any other function, you can compress, expand, and move a log function back and forth and up and down.  For this, our function is y = A logb(B(x-C))+D , where little b is the base of the log, and the Capital B is B in A, B, C and D.  This is a good video which shows all the transformations, starting with an example for only C and D, and then doing an example with all four.

8.3 Logarithm Laws  Goes over the basic properties of logs.  If you don’t want to watch this, they’re summarized in your text on page 394…  The video has examples, though, which is good.

8.4 Logarithmic and Exponential Equations  And this brings us to the end of Chapter 8.  I hope you’ve enjoyed your flight, please wait until the plane has come to a complete stop before you remove your safety belt and remove your items from the overhead compartments.  Thank you and have a nice day.

 9.1 Rational Function Transformations  This takes longer than necessary, so you can probably skip through it.  Goes over the basic rational function, f(x) = 1/x

Here’s a pdf that goes into a bit more detail, but isn’t quite as long-winded… 9.1 Exploring Rational Functions

9.2 Analysing Rational Functions.

No video today.  Just Notes!9-2 Analysing Rational Functions Notes

9.3 – We did it in class!

10.1 Sums and Differences of Functions.

Just some notes today.  10-1 Sums and Differences of Functions

10.2  Products and Quotients of Functions

Again, notes.  Just b/c…10-2 Products and Quotients of Functions

10.3 Composite Functions  Good video from our old friend on how to create and solve composite functions.

Domain is a little tricky – basically, it’s all the x-values in g for which g(x) is in the domain of f… We’ll talk in class…

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