MAT1375 Precalculus Spring2014 MW

Topics:

• Review of Session 8 material:

The following three things are equivalent:

1. c is a root of the polynomial f(x)
2. x-c is a factor of f(x)
3. When you divide f(x) by x-c, the remainder is 0.

Therefore, finding roots is the same process as finding factors of a polynomial, and we have several ways to do things.

To check if x=c is a root of f(x), you can either:

* Substitute c to find f(c) and check to see if f(c) = 0

[Note: you can sometimes use the Table function in your graphing calculator to help with this, but take care in this!]

or

* Divide (preferably synthetically) by x-c to see if the remainder is 0

Usually it’s a good idea to do this, because we may want to know the quotient to find further factors or roots of f(x).

• How a double root appears in a graph and in the factorization of the polynomial.

• What goes into a complete graph of a polynomial, and looking at the output of the graphing calculator.

I am putting together a list of all the things that must appear in a complete graph, and will post it when it is done.

UPDATE: here’s the improved list:

Facts about polynomial graphs:

• The domain of a polynomial is the whole real line

• The y-intercept is the constant term

• For a polynomial of degree n, there are at most n real roots (which are the same as the x-intercepts of its graph) and there are at most n-1 local maxima or minima (turning points).

• The graph of a polynomial is continuous – no breaks or jumps

• The graph of a polynomial does not have any corners

• The graph of a non-constant polynomial does not contain any horizontal line segment

• The end behavior of the graph is determined by the leading term of the polynomial.

We saw in the examples that it is not always possible to get a complete picture of the graph using only one viewing window. Sometimes we need more than one view to get all the details.

• The Rational Roots Theorem – a way to tell whether a rational number is even a possible root of a polynomial f(x).

Basically, the theorem says that a rational number can be a root of f(x) only if its numerator is a factor of the constant term, and its denominator is a factor of the leading coefficient.

• Review of complex numbers

• The fundamental theorem of algebra and its consequences.

The important thing to remember is that the Fundamental Theorem implies that every polynomial of degree n>0 has exactly n roots, if you allow complex numbers as roots and you count each root with its multiplicity. (The multiplicity of a root is the number of times its factor appears in the factorization of f(x), in other words, it is the exponent that goes on that factor.)

Homework:

• Reread and review the definitions, vocabulary, and the examples discussed in class. Make sure that you understand the reasoning that is being used in the examples.

• Do the assigned parts of Exercises 8.3 and 8.4 (note: assigned last time!), Exercise 9.4, and do Exercise 10.2

• No WeBWorK this time.

• Do the Warm-Up for graphs of polynomials (really an “old-up”!) by Tuesday 11 PM. Start early!

• Read ahead Session 10 as time permits. It will help a lot.

The WeBWorK and Warm-Up deadlines from the last class have been extended to Sunday evening 11 PM. I also fixed a bad link to the Warm-Up.

Topics:

• Synthetic division (a short way to do long division when dividing by x-c)

• Polynomial graphs: see below

• Roots, factors, and remainders

Facts about polynomial graphs:

• The domain of a polynomial is the whole real line

• The graph of a polynomial is continuous – no breaks or jumps

• The graph of a polynomial does not have any corners

• For a polynomial of degree n, there are at most n roots (which are the same as the x-intercepts of its graph) and there are at most n-1 local maxima or minima.

• The end behavior of the graph is determined by the leading term of the polynomial.

Homework:

• Reread and review the Observations and the examples discussed in class.

• Finish the assigned problems from Session 8

• In Session 9, do the assigned problems from 9.1, 9.2, and 9.3

• Do the Warm-Up – due tomorrow by 11 PM! extended to Sunday 11 PM (link fixed)

• Do theWeBWorK – a very short assignment – due tomorrow by 11 PM! extended to Sunday 11 PM

See the flyer below…

MathSpecialistHelpNYCCTSpring2014

Topics:

• More on inverse functions: the “round-trip theorem”:

If f and g are one-to-one functions, then they are inverse to each other if and only if both of the following are true:

for all x in the domain of g

AND

for all x in the domain of f.

This theorem provides a way of checking if two functions are inverse to each other.

• The domain of is the range of , and the range of is the domain of .

• The relationship between the graphs: the graph of  is the reflection of the graph of  over the line y=x. (This is because they reverse the roles of y and x.)

• For a function which is not one-to-one: Restricting the domain to an interval where the function is one-to-one, so that we can find an inverse on the restricted domain. Note that when we restrict the domain, we want to make sure that the entire range of the function is covered in the restricted domain.

The example we discussed is . Here is a bit more detail than I gave in class.

If we restrict the domain to the interval , then f(x) is one-to-one on that interval and every value in the range of f is reached for some value of x in the restricted domain. [This would not be true if, for example, we restricted to the open interval . Then f would be one-to-one on the restricted domain, but the value 0 in the range of f would not be in the new range of the restricted f, because no x-value in the new domain gives f(x)=0.]

On that restricted domain, the inverse function is . We can see this by the algebraic method:

(remember that we are taking  when we restrict the domain)

Reverse x and y:  (and now )

Solve for y: 

– We only take the positive branch of the square root, because  due to the restriction on the domain of f(x). So there is a uniquely defined inverse function for f(x) on this restricted domain.

What if we restricted the domain to the interval  ? On this interval f is one-to-one (this is just the left-hand side of the graph) and the entire range of f is covered. But in this case, the inverse function on the restricted domain would be . Why is this so? Because when you try to find the inverse algebraically,

(remember that we are taking  when we restrict the domain)

Reverse x and y:  (and now )

Solve for y: 

– We now only take the negative branch of the square root, because  due to the restriction on the domain of f(x). So there is a uniquely defined inverse function for f(x) on this restricted domain, but it is different from the inverse we get if we restrict the domain as we did the first time.

Moral of the story: there may be more than one choice for how to restrict the domain of a function which is not one-to-one, but different choices will lead to different inverse functions.

• Polynomials: definitions and important vocabulary you should know. (See the list below)

• Long division of polynomials.

Next time we will learn a shortcut efficient method for doing long division of polynomials in certain cases. But there are still times when we will need to use the full long division, so make sure that you practice it!

Vocabulary you need to know for polynomials:

• monomial

• polynomial

• term

• coefficient

• degree (of a term, or of a polynomial)

• constant term – note that the degree of a constant term is defined to be 0.

• root of a polynomial (= zero of the polynomial function = x-intercept of the graph of the polynomial)

• Related to division: dividend, divisor, quotient, remainder

Homework:

• Reread and review the definitions and examples discussed in class. Make sure that you know and can use all the vocabulary correctly!

• Do the assigned parts of Exercises 7.3, 7.4, 8.1, and 8.2

• Take a look at the “Review of functions and graphs” which follows Session 7. (But skip #1.7, which we did not discuss, unless you want to try it – it’s not hard.)

• Do the WeBWorK: due by Sunday 11 PM. Start early and use the Piazza discussion board if you get stuck on any of them (or on the regular homework)!

• Read ahead in Session 8 and then do the Warm-Up for Remainders – due also by 11 PM Sunday. (This one uses a Google Docs form instead of Piazza. Please only submit your answer ONE TIME. You should print or save the receipt that you get when you have submitted it.)

Please note that from now on you are expected to do all the online assignments on time and to inform me promptly if there is any difficulty doing them. I have been very flexible so far, but it’s time to get serious.  I believe we have had time to shake out all the bugs in the system. There will be no more re-assignments or extensions except under very unusual circumstances. So get to it!

(after Test 1)

Topics:

• One-to-one  functions (also called injective functions)

• The horizontal line test for a one-to-one function

• Definition of the inverse function

• Finding a formula for the inverse of a one-to-one function algebraically

Important Notes:

• For a one-to-one function , the inverse function is denoted by . This is read “f inverse of x”. It is not the -1 power! We happen to use the same notation, but when applied to the name of a function it means “inverse” and is not an exponent or power.

• The inverse function reverses the roles of the input and output. Therefore, the domain of  will be the range of  and vice-versa.

• To find the formula of the inverse function, we change x to y and y to x in the formula for the original function, and then solve for y. If we can solve for y (the new y) uniquely in terms of x, then the original function was one-to-one and all is well. If we cannot solve uniquely for y in terms of x, it means that the original function was not one-to-one and so does not have an inverse.

For example, take (which we already know is not one-to-one). If we go ahead and try to find a formula for its inverse, we get this:

(the original function’s formula)

(reversing the roles of x and y)

(I put the term with y on the left, to make it look nicer)

(using the square root principle to solve for y)

So we cannot solve uniquely for y in terms of x after reversing the input and output. This is a reflection of the fact that the original function was not one-to-one.

• As in the example we worked in class, the inverse function will do the opposite (inverse) operations in the reverse order compared to what the original function does.

Homework:

• Reread and review the definitions and the examples worked in class.

• Do the assigned parts of Exercises 7.1 and 7.2

• Make sure that you do the WeBWorK – there is an old one which is extended because there were some problems with it, and also a new, short one. They are due Tuesday (tomorrow!) by 11 PM, so start early!

• Make sure that you do the previously assigned Warm-Up (Warm-Up for Transformations) in Piazza, if you have not already done it. (You can only do it one time.) This is also extended to tomorrow 11 PM because of difficulties that came up in it, but there will be no more extensions!

* Please make sure that you have entered an email address in WeBWorK. I am still getting emails from WeBWorK that I cannot respond to because the students do not have email addresses in WeBWorK. Also, WeBWorK will be used to send out midterm grades later on.

* From now on you should expect that there will be a WeBWorK assignment and a Warm-Up every time (except when there is a Test in the next class meeting), and they will be due by 11 PM the evening before the next class meeting. You should always start early in case there are any difficulties. If you want to discuss how to solve the problems (or homework in general), please use Piazza!

* There is a Piazza app available so you can easily use the Piazza discussion board on your smart phone or iPod touch. See this page for more information.

Topics:

• Composition of functions: Definition 6.5 p. 77. Start here!

• Finding the formula of a composite function

• Using functions defined by tables

• The square root principle (see this post) vs. the definition of square root: what is ? (Came up in Example 6.8(c)) – see fuller discussion below

The square root principle and related things:

The square root principle (as it is sometimes called) says that

if

then .

This is related to two other facts connecting squaring and square roots:

• — This is the definition of the square root of x. ( is the number which, when squared, gives you x.)

• — This is because the radical sign indicates that you should take the non-negative square root, so  is always greater than or equal to 0 no matter what x is.

Many people get the bad idea in their heads that  is the same as x. That is not true as a general rule. It is only true when . Be careful!

Homework:

• Reread/ review the examples. Make sure that you understand how composition works: we go through first one, then the other, of the functions.

• See the homework from last time – especially #5.3 (needed for Test 1!)

• Do the assigned problems from Session 6, but omit 6.1 and 6.2

• No new WeBWorK or Warm-Up today, but I have extended one WeBWorK until tonight 11 PM so if you got cut off you can do it (no further extensions!!!) and the Warm-Up on Piazza is open until the night after Test 1.

• Test 1 Review

Test 1 is scheduled for Monday 24 February, the first 50 minutes or so of class. Please review the course policies: in particular, there are no make-up tests and no extra time for latecomers, so please be early!

I handed out review Self-Tests in class, and I will also post them to Piazza. You can use Piazza to discuss these problems or to tell me if you find any typos!

The answers follow, after the fold. Please use the Piazza Discussion Board to discuss these!

Continue reading →

Topics:

• Basic graphs: the graphs of the following functions are essential to know:

Important feature: this graph has an inflection point (the s-curve) at (0,0)

Important feature: the domain is only the interval

Important feature: the graph approaches the lines x=0 and y=0 (the two coordinate axes). These lines are called asymptotes. Notice that x=0 is not in the domain.

• Transformations of graphs:

vertical translation (shift) – add a number to the output (value of the function)

horizontal translation (shift) – add a number to the input

vertical stretching or squeezing – multiply the output by a positive number

horizontal stretching or squeezing – multiply the input by a positive number

reflection in the x-axis – multiply the output by -1

reflection in the y-axis – multiply the input by -1

• Even and odd functions:

Basically, even functions have graphs which have the same symmetry as the graph of and odd functions have graphs which have the same symmetry as the graph of . The algebraic tests are tests for those symmetries.

Algebraic tests:

A function is even if and only if for all x in its domain.

A function is odd if and only if for all x in its domain.

Homework:

• Reread/ review all the examples, making sure that you see how the transformations change the graphs (and why!) and how the algebraic test for an even or odd function checks the symmetry. You can use Piazza to discuss if you have questions.

• Do the assigned problems from Session 5

• No WeBWorK or Warm-Up today because we have class tomorrow!

• Don’t forget Test 1 Review

Topics:

• Careful use of = signs in doing computation of difference quotient (or anything else): more similar usage comments are below.

• Using the graphing calculator: see the handouts for topics. In this context we encountered Wheeler’s first moral principle: never solve a problem until you know the answer. (We saw that you need to know more or less what the graph looks like in order to be able to interpret what the graphing calculator was showing. But the principle is generally useful.)

• The square root principle: see below

Usage comments:

• Don’t use arrows in place of equals signs. The double arrow is used between statements (for example, two equations) and it means “if the first one is true, the second one is also true”. You should avoid using it unless absolutely necessary: instead, put each successive equation on a separate line.

• In general, leave a lot of space in your work. It makes it easier to read and understand, and it makes it easier to make corrections!

• When solving an equation of the form (something), we use the Square Root Principle:

If , then

$latex x = \pm\sqrt{c}

Note: there is no “missing step”. We do not “take the square root of both sides”. What is wrong with doing that has to do with the fact that is NOT necessarily x. (And yes, I know that some teachers tell you to “square root both sides”, ugh, but it is still wrong, and will cause you trouble later on.)

Here is a video that explains it correctly. (You’ll have to sit through an annoying ad first, but the video is good and is totally mathematically correct.)

If you write the incorrect version, it is an error (that’s the main consideration!) and will cause you to have fewer points on that problem (if that’s all you care about). So correct this in your thinking if necessary!

Homework:

• Review the examples discussed in class. Especially make sure that you understand the usage notes above, and the square root principle. Practice on the examples in the textbook, making sure that the things on each side of an = sign are actually equal to each other.

• Practice practice practice with your calculator, remembering to examine your results against what else you know as a check

• Do the assigned exercises from Session 4

• Do the WeBWorK assignment (due by Tuesday the 18th 11 PM) – start early!

• Join Piazza (if you have not already done so) and join the MW group in Piazza – and use it if you have any questions about the homework problems! If you responded to the first Warm_Up (Exercise 1.1) you should have received an email inviting you to join the Piazza for this course

• Do the Warm-Up in Piazza (also due by Tuesday the 18th 11 PM)