Susan Cantey has taught mathematics for over 30 years. This is the second of many CD's to come inspired by her love of music and the desire to help her students enjoy their mathematical studies. The art work on the CD cover was created by one of Susan's PreCalculus students: Gwen Wood, c/o 2010. Gwen is a good math student and a wonderful artist. To see her students' videos, type MrsCantey into the search box at You Tube. Lyrics: Never Never Never Susan Cantey © 2008 I will never never never never never never never ever Distribute an exponent… I will never never never never never never never ever Distribute an exponent… I will never never never never never never never ever Distribute an exponent… I will think about what’s meant by an exponent And follow all of their rules. I will never never never never never never never never ever distribute a square root… (repeat) I would never abuse the Distributive Law Which doesn’t work with the square root at all. I will never never never never never never never never ever cancel part of a sum… (repeat) It should be a sin to do that kind of thing, I will factor first ‘cause that’s how it’s done. (repeat verse) I will never never never never never never ever Miscopy messy sloppy work… (repeat) Mistakes will be erased; I’ll work neatly down the page, Clearly showing every step. (repeat verse) I will never never never never never never ever distribute an exponent… I will never never never never never never never never Ever distribute a square root… I will never never never never never never never never Ever cancel part of a sum… I will never miscopy messy sloppy work, Yeah, these are the things I’ll never do! What’s a Function? Susan Cantey © 2008 What’s a function? It’s a number machine, It takes a number From a set called the domain, And with its rule Oh, it makes a change, Spits it out Into a set called the range. If “x” is a number inside the domain, It doesn’t care about being changed, Takes on any value of its own free will It’s an independent variable. But if “y” is a number within the range, Its very essence depends upon the change, Without an x-value, it’s at a stand still, It’s a dependent variable. (repeat from start) Domain Susan Cantey © 2008 Domain…you gotta know Domain…watch out for zeros Domain…in denominators Domain…that’s a no no Domain…negative numbers Domain…never go under Domain…square roots or radicals Domain…with even indices Eliminate the x’s that exacerbate the function Mathematically criminal demand your attention Don’t let illegal x’s cross the border out of order ‘Cause results of all manipulations gotta be real numbers. CHORUS Zero at the bottom puts a hole in your domain If that factor cancels out then the range will see the same But a factor never cancelled that always must remain Makes a vertically asymptotic mess out of the range CHORUS Put this in your brain No need to go insane Keep on singing this refrain When you need the domain CHORUS Home, Home on the Range Susan Cantey © 2008 There is a home, where the y-values roam, And students find themselves strained, Where often is heard a discouraging word, When all they can find’s the domain. Chorus: Home, home on the range, Where the y-values must stay, It may be preferred To use a graph and observe The part of the y-axis in play. Often at night, students try to do right, Making tables of x’s and y’s, The range should be clear, but their eyes start to tear, And trouble begins to arise. (Chorus) I’d rather be lying on a beach On some tropical sand, But I have to face reality, Find the range the best way I can. If you don’t have the graph, it’s a difficult task, If it was easy then anyone could do it, You might need calculus to find mins and maximums, If a table becomes insufficient. (Chorus) Composite Function Susan Cantey © 2008 Composite function, composite function, When a function takes the place of “x”, In a function, to make a function More complex than the rest, Ah…. If f(x) is one over “x” then f(g) is one over “g”, If g(x) is “x” squared then g(f) is “f” squared, Ah…. Interlude If f(x) is three “x” plus sine “x” Then f(g) is three “g” plus sign “g”, And if g(x) is the square root of “x” cubed Then g(f) is the square root of “f” cubed, Ah… (repeat first verse) Odd and Even Functions Susan Cantey © 2008 Odd and even functions, I think you’ll agree, Have absolutely no ambiguity, So here is the skinny, here is the test, Replace x with a negative x, If the function stays the same, Then it’s even, yeah, that’s its name, If it’s exactly the opposite of The original function, Then it is odd. Even functions are symmetric to the y-axis, You can reflect it, Odd functions are symmetric to the origin, There’s only one thing, That’s just a guess, Ain’t proved nothing yet. Repeat from start f(x), f(-x), compare the two, that’s what you do. Even functions are symmetric to the y-axis, You can reflect it, Odd functions are symmetric to the origin, There’s only one thing, That’s just a guess, Ain’t proved nothing yet. Transformation Tango Susan Cantey © 2008 CHORUS: Take me, shake me, Squeeze me, stretch me, Flip me, slide me, That’s what gets me, Transformation Tango, Here I go… Add to the x, move me to the left, Add to the rule, I’m up like a fool, Subtract from the x, move me to the right, Subtract from the rule and down I slide! CHORUS Multiply, I’ll intensify, Watch me grow, maybe fast, maybe slow, Negative sign flips me every time, In front – over x, inside – over y… CHORUS Repeat from start… Big Switcheroo (Inverses) Susan Cantey © 2008 CHORUS: One to one, happy function, Passes both of the line tests, So it knows it has an inverse, Monotonic, a new y for every x, One to one, monogamous Plus one, minus one, times two, divide by two, Squaring, square root, the inverse will undo, It’s just a big switcheroo. (repeat chorus) Y equals x, reflect the graph, the image made is half the task, Take the rule specified, switch the old x and y It’s just a big switcheroo. (repeat chorus) BRIDGE: The inverse has rearranged; domain and range will be exchanged… It’s just a big switcheroo! (repeat chorus) Polynomials Susan Cantey © 2008 Polynomial, “a” plus “a” one “x” Plus “a” two “x” squared plus “a” three “x” cubed, And so on and so on Till you stop at its degree. When the degree is only one, Well its graph is not a lot of fun, Has no turns, it simply travels straight, At a constant sloping rate, With degree of two, You know what to do, Parabola with a shape that looks just like a U, Degree three we have a cubic With two turns just like a snake. If the degree is four You can have one or three turns, It can be like a W Or just a steeper flatter U. There’s a pattern here, Listen, do not fear, All the information that you’ll ever, ever need, Is hidden in the highest power, What we call its degree. Oh, the number of possibilities, For turns depends on that degree, Just subtract one from the highest power, That’s the most turns the function is allowed, Polynomial doesn’t have to use All the turn available, They might reduce by twos, If you’re missing turns that might indicate Double or imaginary roots. Now you know everything you need To graph a polynomial, So please go now henceforth And draw them smooth, unbroken and well. Polynomial, polynomial, Poly, poly, poly, poly, polynomial, Poly, poly, polynomial, Poly, poly, polynomial, N minus one turns maximum, Could go down by twos…. Discontinuity © 2009 Susan Cantey A discontinuity is a break in the graph, There are three main types of discontinuities, Three types of discontinuities, Type one is a hole, removable, Just a single missing point, Type two is a step or a jump Where you must lift your pencil, To a different y-value, But the biggest discontinuity, The biggest discontinuity, Is infinite, yes infinite, Is infinite, yes infinite, Is infinite, yes infinite, It goes on forever, It’s a vertical asymptote, It’s a vertical asymptote, The graph goes up or down forever, Forever, forever, Now you know the three main types…of discontinuities, So get ready to lift your pencil up… When there is a discontinuity. Rational Functions Susan Cantey © 2008 A rational function is a ratio Of two polynomials, The denominator tells you where the asymptotes are, The numerator gives you the ze-e-e-roes, Every little tiny piece Of the graph that you complete, Looks like the factor that you found in the function Near the asymptote or zero where you happen to be drawing… Hi ho, hi ho, Where do you start? Where do you go? Oh what fun you’re gonna know When poly’s form a ratio, uh oh, When poly’s form a ratio. End behavior’s what you need At the start so you will see The y value that you get when x is big, It’s not as complicated as you might think, Take a look at the leading terms, Reduce them down and ask yourself “When x is really big, what happens to y?” That’s where you’ve gotta start drawing on the right… Hi, ho, hi, ho, Where do you start? Where do you go? Oh what fun you’re gonna know When poly’s form a ratio, uh oh, When poly’s form a ratio. (repeat) It’s Exponential Susan Cantey © 2008 From the bank, ya get yer compound interest, Your balance will be equal to this, Original deposit times the quantity, One plus “r” over “n” to the “n” “t”, “r” is the annual interest rate And “n” is the number of times they calculate, “t” is the number of years ya gotta wait, If ya got a lotta money, well that’s just great. Who’s got the power? (It’s exponential!) Variable power? (It’s exponential!) Growth and decay! (Hey! Hey! Hey!) Looks like a ski slope… (Yo! Yo! Yo! Yo!) 5, 4, 3 or 2 to the “x”, “e” to the “x” is the very best…. I’m explainin’ the equation as atoms decay, “A” “e” to the “k” “x” is O.K., The half life is “l” “n” of one half over “k”, Radioactive material tries to go away, But if k is positive then ya got growth, Population, no attrition, that’s the way it goes, Can’t go on forever ‘cept mathematically, So the model only works initially. The basic function that you’ve got Is “a” times “b” to the “x”…not a lot Of information needed, y’all, Just sketch this thing called an exponential, If “b” is big then the graph’s up hill, “b” less than one and we’ll just chill, Draw that graph on down to the right, Asymptotic x-axis, yeah, we’re tight! Logarithms Susan Cantey © 2008 In math there’s something called a logarithm, No need to worry when you see ‘em, They are easy if you think of Logarithms as fancy exponents, Log base “a” of “b” is “c”, If “a” to the power of “c” equals “b”, In other words, “c” is the power you need, To change that base “a” into “b”. Log base two of eight is three, ‘Cause three is the exponent that you need To change the two into an eight, Here’s some more examples to demonstrate: Log base three of nine is two, ‘Cause to change three into nine, squaring’s what you do, Log base ten of a million is six, Log base two of a half is negative one ‘cause it flips. There are rules you’ll have to use, Log of “b” to the “n” it’s true, Equals “n” log “b” you see, Log of “ab” equals log “a” plus log “b” Log of one in any base, Will equal zero in every case, Log base “b” of “b” is one, Negative bases, no there’s none, If there’s no base then assume it’s ten, Base e is the best and its abbreviation Is “l” “n” and this base is The one we use in calculus. Power, power, logarithmic power…. Log of “a” divided by “b”, Equals log “a” minus log “b”, Change log base “a” of “x” to a different base, Log base “b” of “x” over log base “b” of “a”… (repeat first verse) Radians are Rad Susan Cantey © 2008 In ancient Babylon someone drew a hexagon And broke each section into sixty pieces, It makes no sense today measuring angles in that way, It’s a supercilious system old as Croesus, Why do we keep using three sixty degrees? It’s a mathematical mystery. Chorus: Radians are rad; degrees make me sad, Oh those Babylonians, they really make me mad, Rads are OK; degrees are passé, Angles based on six times sixty need to go away! Makes a lot more sense to use circumference, In every case that’s always two pi radii, Pi is really swell, but if that don’t ring your bell, Approximate with three point one four one five nine, Now the circle measures six point two eight, Metric measurement is the rage. (repeat the chorus) Out in the universe there might be other creatures Who may not have ten fingers, that’s very true, Certainly base sixty will not be common either, But pi will always be used! (repeat the chorus) I Love That Circle Susan Cantey © 2008 Cosine is the x-value on the Unit Circle, Sine is the y-value on that circle too, Beautiful circle; beautiful circle, Basis of trigonometry. I love that circle with radius one, Beautiful circle looks like the sun, Where all of trigonometry radiates from. Tangent is the fraction: y over x, x over y is cotangent; I will never forget Beautiful circle; beautiful circle, Basis of trigonometry. (repeat chorus) Cosecant theta, cosecant: one over y, Secant theta, secant: one over x, makes me cry, Beautiful circle; beautiful circle, Basis of trigonometry. (repeat chorus) The Unit Circle Workout Susan Cantey © 2008 Put your hands in the air; now move them to the right, That’s the positive x-axis; we’re gonna move counter clockwise. Are you ready for the Unit Circle Workout? Zero, pi over six, pi over four, pi over three, Pi over two, two pi over three, three pi over four, five pi over six, Pi, seven pi over six, five pi over four, four pi over three, Three pi over two, five pi over three, seven pi over four, eleven pi over six, Two pi, two pi, two pi…. Put your hands back in the air, put ‘em back to the right, Still going counter clockwise, doing sine this time. Zero, one half, root two over two, root three over two, One, root three over two, root two over two one half, Zero, negative one half, negative root two over two, negative root three over two, Negative one, negative root three over two, negative root two over two, negative one half, Zero, zero, zero. One, root three over two, root two over two, one half, Zero, negative one half, negative root two over two, negative root three over two, Negative one, negative root three over two, negative root two over two, negative one half, Zero, one half, root two over two, root three over two, one, one, one, we’re done. This is the last one; we’re gonna do it double time, Arms out to the side; gonna shoot to the sky, Tangent…double time! Zero, one over root three, one, root three, Undefined, negative root three, negative one, negative one over root three, Zero, one over root three, one, root three, Undefined, negative root three, negative one, negative one over root three, Zero, zero, zero, zero, now we’re done Trigonometric Graphs Susan Cantey © 2008 CHORUS: In trigonometry, all of the graphs repeat, One cycle and that is how the rest of the graph will be, The period’s two pi, The Unit Circle is why, Except for the tangent and cotangent Which are only half that size. Oh the sine curve starts at the origin, Makes a frowny face, then a happy face, And the cosine curve starts where y is one, Half a frown, a smile then upwards again. CHORUS Oh the tangent curve has vertical asymptotes, Plus or minus one half pi, Then it snakes uphill through the origin, And cotangent goes down from zero to pi. CHORUS Trigonometric Identities Susan Cantey © 2008 There are so many identities, Only memorize the ones you need, Start with the quotients and then progress Pythagorean and all the rest, Secant is one over cosine x, Cosecant is one over sine x, Tangent is the sine over the cosine, Cotangent is cosine over the sine. Sine squared x plus cosine squared is one, Move something to the other side for fun, Or divide by sine or cosine squared, Tangent squared plus one is secant squared. There are so many identities, Only memorize the ones you need, Half angle double angle, then all that’s left, Are the ones called the sum and difference, Sine squared x, don’t forget Is one half minus one half cosine two x, Cosine squared is the same thing too, Except with a plus sign, yes, it’s true. Sine two A is two cosine A sine A, Cosine two A, cosine squared A minus sine squared A, Tangent two A is 2 tangent A Over one minus tangent squared, hey! Sine of the quantity A plus B is Sine A cosine B plus sine B cosine A, If it’s a minus, change both the plus signs, It’s quite simple by design, Cosine of A plus B is Cosine A cosine B minus sine A sine B, Keep the like ones together naturally, With minus use pluses, the signs will disagree. (short solo) There are so many identities, Only memorize the ones you need, Start with the quotient and then progress, Pythagorean and all the rest. Stay in the Principle Domain Susan Cantey © 2009 Inverse trigonometric functions give you angle measures, When you use the trig values that you’ve learned in school, It’s important that you pay attention to the special portion, the piece that plays the part, Only invert a tiny little section of the trigonometric graph. CHORUS: Stay in the principle domain, yeah, yeah, yeah, Stay in the principle domain, It’s the inverse trigonometric way, You’ve got to stay, stay, stay in the principle domain. Sine and tangent inverse gotta stay in quadrants one and four, Cosine’s in between zero and pi, you get positives and negatives that way, People disagree on the other three, so we’ll leave them on the shelf, Just need to know two principle domains and nothing else. CHORUS Negative pi over two to pi over two (sine and tangent), Or zero to pi (cosine) CHORUS Polar Graphs Susan Cantey © 2009 Imagine you’re standing on the origin facing zero degrees, Walk along that positive axis until r units are reached, Now rotate along a circular path counter clockwise theta degrees, Put your point right there in that spot – polar coordinates are easy. Polar Graphs, polar graphs, cardioids and roses, Lemniscates, limacons, spirals, conics and so on. (repeat) “r" equals “a” cosine or sine “n” theta will make a beautiful rose, “n” petals if “n” is odd; double “n” if “n” is even of course, A limacon will form, if you have “a” plus or minus “b” sine or cosine theta, You’ll get a loop if “a” is less than “b” but the loop shrinks as “a” grows greater. (repeat chorus) A cardioid is just a limacon where “a” and “b” are equal, If “a” is bigger than “b”, the limacon will be dimpled, When “a” is twice as big as “b”, no dimple will be there, Lastly, the lemniscate that looks like a bow tie, “r” squared equals “a” squared sine or cosine two theta. (repeat chorus twice) Linear Systems Susan Cantey © 2009 The first equation my teacher gave to me was a linear equation in “x”. The second was a system, that’s what she gave me next, Two equations in “y”and “x”, The third kind of system my teacher gave to me Had three equations, three variables using “x”, “y” and “z”. The fourth kind of system my teacher gave to me Had four equations, four variables, “w”, as well as “x”, “y” and “z”. The fifth kind of system my teacher gave to me: five variables! That’s when I reached for my calculator, Doing this one by hand is extreme. The sixth kind of system my teacher gave to me I got a row of zeros; what does that mean? The system is dependent, there’s more than one solution, They are infinite. The seventh kind of system my teacher gave to me Had six zeros followed by a one; what does that mean? The system’s inconsistent, and if that’s what you get, the answer’s the empty set. The eighth kind of system had eight variables, Use substitution or combinations; I said forget it: There is no way! Unless you take me back to fewer variables, there are not enough hours in my day. The ninth kind of system my teacher gave to me Was very easy, x – y is 7, x + y is 3, add them together, 2x = 10, “x” is five and “y” is negative two, at last that’s something I can do. The tenth kind of system my teacher gave to me, 2x + y = 5, 3x + 4y = 10, I used substitution, y = 5 – 2x, now put that in the other one, Solve for “x” and soon I’ll be done. Now that gives me 3x + 20 – 8x = 10, -5x is -10, divide by -5, and then x = 2, y = 1, now we’re done, so I’ll say so long to the systems in this song. Conic Sections Susan Cantey © 2008 These are the most common conic sections, Circles, ellipses, hyperbolas, parabolas, Circles are easy “x” squared plus “y” squared is Equal to the radius squared, If the center’s at the origin. An ellipse is a circle with a variable radius, “a” is the biggest one and “b” is the smallest, “x” squared over “a” squared Plus “y” squared over “b” squared equals one, That’s a short and wide one, Switch the “a” and “b” get a tall and thin one, “a” squared minus “b” squared equals “c” squared, Which locates the focus. Hyperbolas are like inside out ellipses, And their equations, instead of plus, have a minus, Now “c” ‘s the biggest value; “a” is the one that’s first, Leading variable tells you where to put the vertex. Parabolas are different and that’s no lie, The U-shaped ones are functions, “x” squared equals four “cy”, “c” ‘s the distance from vertex to focus And also from vertex to directrix, If you switch the “x” and “y” it’s no wonder, The parabola will be on its side, “4c” when you’re done Is the width of the latus rectum. These are the most common conic sections, Circles, ellipses, hyperbolas, parabolas, La….la…la… Summation Notation Susan Cantey © 2008 Summation notation, it’s an invitation To add some numbers up, The number at the bottom and the number at the top Tell you where to start and stop, Put each number in the formula one by one, Add them all together to get the sum, Summation notation, it’s an invitation To add some numbers up. I don’t know why it’s scary, Just a capital E, Sigma is Greek… But not as strange as it seems. (repeat) Sequences and Series Susan Cantey © 2008 The counting or whole numbers are put into a rule To create an ordered sequence and then, Instead of “f” of “x”, the notation’s more complex, Instead of “f”, it’s “a”; and for “x” we use an “n”. Separate the numbers in a sequence with commas, Put the rule inside curly braces if you like, Separate the numbers in a series with plus signs, Sigma drawn before the rule makes your work look right. If the rule is to add a number it’s an arithmetic sequence, The number that you add is called the common difference, “a” sub “n” equals “a” sub “k” plus “n” minus “k” times “d”, “S” sub “n” equals “n” over two times the quantity of “a” sub one plus “a” sub “n”. 1, 2, 3, 4, 5, 6, 7… 1, 4, 9, 16, 25, 36… 4, 2, 1, , , … 1, 3, 1, 3, 1, 3, 1… The first one was “a” sub “n” equals “n”, The second one was merely “n” squared, The third one was “a” sub “n” is eight over two to the “n”, The last one is two plus negative one to the “n”. If the rule is to multiply then the series is geometric, The factor that you multiply by is the common ratio, yeah, that’s it, “a” sub “n” equals “a” sub “k” times “r” to the “n” minus “k”, “S” sub “n” equals “a” sub one times one minus “r” to the “n”, hey, Over one minus “r”. Given geometric series, absolute value of “r” less than one, First term over one minus “r” is the infinite sum, There’s no reason to fear the notation we use, Just familiarize yourself and soon you’ll feel good… Permutations and Combinations Susan Cantey © 2008 How many ways can you line up ten boys? How many ways can you line up ten books? How many ways can you line up ten toys? Well, here’s the way it looks. According to the counting principle, It’s got to be ten factorial! Chorus: If order makes a difference, use permutations, If order makes NO difference, use combinations, That’s the difference in the computation! How many different four number codes, With ten numbers to choose from? That’s permutation ten “P” four I’m told, ‘Cause the order’s important, But when choosing a committee, That’s combinations ‘cause it’s order free. (repeat chorus) Instrumental break (repeat chorus) Around a table in a circular arrangement, “n” minus one factorial, If it’s something you can turn over, Divide by two and all is well, What about the letters in a word? Divide by factorials for any repeated. (repeat chorus) Probability Susan Cantey © 2009 Probability, What are the chances? Between zero and one will be All of your answers, When all else fails, List the sample space…sample space. Every single member, Must be equally likely, The total number, Is the denominator, The number of successes are, The numerator…yeah, yeah, yeah. Probability “one” is a sure thing, Zero means there’s no way, Fifty percent, maybe no, maybe yes, The chance is exactly the same. When all else fails, List the sample space…sample space. Independent events Do not affect each other, But mutually exclusive events Cannot happen together, “Or”…adding’s what you try, “And” means you multiply…you gotta say it right.